A001055 -id:A001055

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A045782

Number of factorizations of n for some n (image of A001055).

+20
33

1, 2, 3, 4, 5, 7, 9, 11, 12, 15, 16, 19, 21, 22, 26, 29, 30, 31, 36, 38, 42, 45, 47, 52, 56, 57, 64, 66, 67, 74, 77, 92, 97, 98, 101, 105, 109, 118, 135, 137, 139, 141, 162, 165, 171, 176, 181, 189, 195, 198, 203, 212, 231, 249, 250, 254, 257, 267, 269, 272, 289

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also the image of A318284. - Gus Wiseman, Jan 11 2020

LINKS

Table of n, a(n) for n=1..61.

Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.

FORMULA

The Luca et al. paper shows that the number of terms with a(n) <= x is x^{ O( log log log x / log log x )}. - N. J. A. Sloane, Jun 12 2009

MATHEMATICA

terms = 61; m0 = 10^5; dm = 10^4;

f[1, _] = 1; f[n_, k_] := f[n, k] = Sum[f[n/d, d], {d, Select[Divisors[n], 1 < # <= k &]}];

Clear[seq]; seq[m_] := seq[m] = Sort[Tally[Table[f[n, n], {n, 1, m}]][[All, 1]]][[1 ;; terms]]; seq[m = m0]; seq[m += dm]; While[Print[m]; seq[m] != seq[m - dm], m += dm];

seq[m] (* Jean-François Alcover, Oct 04 2018 *)

CROSSREFS

Factorizations are A001055 with image this sequence and complement A330976.

Strict factorizations are A045778 with image A045779 and complement A330975.

The least number with exactly a(n) factorizations is A045783(n).

The least number with exactly n factorizations is A330973(n).

Cf. A002033, A007716, A033833, A318284, A325238, A330935, A330936, A330977, A330989, A330991, A330992, A330997.

KEYWORD

nonn

AUTHOR

David W. Wilson

EXTENSIONS

Name edited by Gus Wiseman, Jan 11 2020

STATUS

approved

A035310

Let f(n) = number of ways to factor n = A001055(n); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

+20
21

1, 4, 12, 47, 170, 750, 3255, 16010, 81199, 448156, 2579626, 15913058, 102488024, 698976419, 4976098729, 37195337408, 289517846210, 2352125666883, 19841666995265, 173888579505200, 1577888354510786, 14820132616197925, 143746389756336173, 1438846957477988926

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Ways of partitioning an n-multiset with multiplicities some partition of n.

Number of multiset partitions of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities. The (weakly) normal version is A255906. - Gus Wiseman, Dec 31 2019

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..50

EXAMPLE

a(3) = 12 because there are 3 terms in A025487 with 3 factors, namely 8, 12, 30; and f(8)=3, f(12)=4, f(30)=5 and 3+4+5 = 12.

From Gus Wiseman, Dec 31 2019: (Start)

The a(1) = 1 through a(3) = 12 multiset partitions of strongly normal multisets:

{{1},{1}} {{1,2,3}}

{{1},{2}} {{1},{1,1}}

{{1},{1,2}}

{{1},{2,3}}

{{2},{1,1}}

{{2},{1,3}}

{{3},{1,2}}

{{1},{1},{1}}

{{1},{1},{2}}

{{1},{2},{3}}

(End)

MAPLE

with(numtheory):

g:= proc(n, k) option remember;

`if`(n>k, 0, 1) +`if`(isprime(n), 0,

add(`if`(d>k, 0, g(n/d, d)), d=divisors(n) minus {1, n}))

end:

b:= proc(n, i, l)

`if`(n=0, g(mul(ithprime(t)^l[t], t=1..nops(l))$2),

`if`(i<1, 0, add(b(n-i*j, i-1, [l[], i$j]), j=0..n/i)))

end:

a:= n-> b(n$2, []):

seq(a(n), n=1..10); # Alois P. Heinz, May 26 2013

MATHEMATICA

g[n_, k_] := g[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d > k, 0, g[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; b[n_, i_, l_] := If[n == 0, g[p = Product[Prime[t]^l[[t]], {t, 1, Length[l]}], p], If[i < 1, 0, Sum[b[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_] := b[n, n, {}]; Table[Print[an = a[n]]; an, {n, 1, 13}] (* Jean-François Alcover, Dec 12 2013, after Alois P. Heinz *)

PROG

(Python)

from sympy.core.cache import cacheit

from sympy import divisors, isprime, prime

from operator import mul

@cacheit

def g(n, k):

return (0 if n > k else 1) + (0 if isprime(n) else sum(g(n//d, d) for d in divisors(n)[1:-1] if d <= k))

@cacheit

def b(n, i, l):

if n==0:

p = reduce(mul, (prime(t + 1)**l[t] for t in range(len(l))))

return g(p, p)

else:

return 0 if i<1 else sum([b(n - i*j, i - 1, l + [i]*j) for j in range(n//i + 1)])

def a(n):

return b(n, n, [])

for n in range(1, 11): print(a(n)) # Indranil Ghosh, Aug 19 2017, after Maple code

(PARI)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)}

seq(n)={my(s=0); forpart(p=n, s+=D(p, n)); s} \\ Andrew Howroyd, Dec 30 2020

CROSSREFS

Cf. A025487, A000041, A000110, A035098, A080688.

Sequence A035341 counts the ordered cases. Tables A093936 and A095705 distribute the values; e.g. 81199 = 30 + 536 + 3036 + 6181 + 10726 + 11913 + 14548 + 13082 + 21147.

Cf. A035341, A093936, A095705.

Row sums of A317449.

The uniform case is A317584.

The case with empty intersection is A317755.

The strict case is A317775.

The constant case is A047968.

The set-system case is A318402.

The case of strict parts is A330783.

Multiset partitions of integer partitions are A001970.

Unlabeled multiset partitions are A007716.

Cf. A001055, A255906, A269134, A316652, A317654, A318284, A330467, A330471, A330475, A330675.

KEYWORD

nonn,nice

AUTHOR

Alford Arnold

EXTENSIONS

More terms from Erich Friedman.

81199 from Alford Arnold, Mar 04 2008

a(10) from Alford Arnold, Mar 31 2008

a(10) corrected by Alford Arnold, Aug 07 2008

a(11)-a(13) from Alois P. Heinz, May 26 2013

a(14) from Alois P. Heinz, Sep 27 2014

a(15) from Alois P. Heinz, Jan 10 2015

Terms a(16) and beyond from Andrew Howroyd, Dec 30 2020

STATUS

approved

A330991

Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

+20
17

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

In short, A001055(a(n)) belongs to A000040.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

EXAMPLE

Factorizations of selected terms:

(4) (8) (16) (24) (60) (96)

(2*2) (2*4) (2*8) (3*8) (2*30) (2*48)

(2*2*2) (4*4) (4*6) (3*20) (3*32)

(2*2*4) (2*12) (4*15) (4*24)

(2*2*2*2) (2*2*6) (5*12) (6*16)

(2*3*4) (6*10) (8*12)

(2*2*2*3) (2*5*6) (2*6*8)

(3*4*5) (3*4*8)

(2*2*15) (4*4*6)

(2*3*10) (2*2*24)

(2*2*3*5) (2*3*16)

(2*4*12)

(2*2*3*8)

(2*2*4*6)

(2*3*4*4)

(2*2*2*12)

(2*2*2*2*6)

(2*2*2*3*4)

(2*2*2*2*2*3)

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Select[Range[100], PrimeQ[Length[facs[#]]]&]

CROSSREFS

Factorizations are A001055, with image A045782, with complement A330976.

Numbers whose number of strict integer partitions is prime are A035359.

Numbers whose number of integer partitions is prime are A046063.

Numbers whose number of set partitions is prime are A051130.

Numbers whose number of factorizations is a power of 2 are A330977.

The least number with prime(n) factorizations is A330992(n).

Cf. A001222, A033833, A045783, A181819, A181821, A326622, A330972, A330973, A330993.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 07 2020

STATUS

approved

A330977

Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

+20
13

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The complement starts: 8, 16, 24, 27, 30, 32, 36, 40.

LINKS

Table of n, a(n) for n=1..67.

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

EXAMPLE

Factorizations of n = 1, 4, 12, 72:

() (4) (12) (72)

(2*2) (2*6) (8*9)

(3*4) (2*36)

(2*2*3) (3*24)

(4*18)

(6*12)

(2*4*9)

(2*6*6)

(3*3*8)

(3*4*6)

(2*2*18)

(2*3*12)

(2*2*2*9)

(2*2*3*6)

(2*3*3*4)

(2*2*2*3*3)

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Select[Range[100], IntegerQ[Log[2, Length[facs[#]]]]&]

CROSSREFS

The same for strict integer partitions is A331022.

Factorizations are A001055, with image A045782.

The least number with exactly n factorizations is A330973(n).

The least number with exactly 2^n factorizations is A330989(n).

Numbers whose inverse prime shadow belongs to this sequence are A330990.

Numbers with a prime number of factorizations are A330991.

Cf. A002033, A033833, A045778, A045783, A317145, A326028, A330972, A330976.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 07 2020

STATUS

approved

A080688

Resort the index of A064553 using A080444 and maintaining ascending order within each grouping: seen as a triangle read by rows, the n-th row contains the A001055(n) numbers m with A064553(m)=n.

+20
11

1, 2, 3, 4, 5, 7, 6, 11, 13, 8, 10, 17, 9, 19, 14, 23, 29, 12, 15, 22, 31, 37, 26, 41, 21, 43, 16, 20, 25, 34, 47, 53, 18, 33, 38, 59, 61, 28, 35, 46, 67, 39, 71, 58, 73, 79, 24, 30, 44, 51, 55, 62, 83, 49, 89, 74, 97, 27, 57, 101, 52, 65, 82

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The number 12 can be written as 3*2*2, 4*3, 6*2 and 12 corresponding to each of the four values (12,15,22,31) in the example. Note that A001055(12) = 4. Since A001055(n) depends only on the least prime signature, the values 1,2,4,6,8,12,16,24,30,32,36,... A025487 are of special interest when counting multisets. (see for example, A035310 and A035341).

A064553(T(n,k)) = A080444(n,k) = n for k=1..A001055(n); T(n,1) = A064554(n); T(n,A001055(n)) = A064554(n). - Reinhard Zumkeller, Oct 01 2012

Row n is the sorted list of shifted Heinz numbers of factorizations of n into factors > 1, where the shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1). - Gus Wiseman, Sep 05 2018

LINKS

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

EXAMPLE

a(18),a(19),a(20) and a(21) are 12,15,22 and 31 because A064553(12,15,22,31) = (12,12,12,12) similarly, A064553(36,45,66,76,93,95,118,121,149) = (36,36,36,36,36,36,36,36,36)

From Gus Wiseman, Sep 05 2018: (Start)

Triangle begins:

4 5

6 11

8 10 17

9 19

14 23

12 15 22 31

26 41

21 43

16 20 25 34 47

Corresponding triangle of factorizations begins:

(),

(2),

(3),

(2*2), (4),

(5),

(2*3), (6),

(7),

(2*2*2), (2*4), (8),

(3*3), (9),

(2*5), (10),

(11),

(2*2*3), (3*4), (2*6), (12).

(End)

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];

Table[Sort[Table[Times@@Prime/@(f-1), {f, facs[n]}]], {n, 20}] (* Gus Wiseman, Sep 05 2018 *)

PROG

(Haskell)

a080688 n k = a080688_row n !! (k-1)

a080688_row n = map (+ 1) $ take (a001055 n) $

elemIndices n $ map fromInteger a064553_list

a080688_tabl = map a080688_row [1..]

a080688_list = concat a080688_tabl

-- Reinhard Zumkeller, Oct 01 2012

CROSSREFS

Cf. A001055, A025487, A035310, A035341, A064553, A080444.

Cf. A007716, A056239, A162247, A215366, A275024, A317144, A317145, A318871.

KEYWORD

easy,nonn,tabf

AUTHOR

Alford Arnold, Mar 23 2003

EXTENSIONS

More terms from Sean A. Irvine, Oct 05 2011

Keyword tabf added and definition complemented accordingly by Reinhard Zumkeller, Oct 01 2012

STATUS

approved

A190938

Parity of the multiplicative partition function A001055.

+20
10

1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

Suggested by the title of a talk by Paul Pollack in the program for the 2011 Illinois Number Theory Conference.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Paul Pollack, The parity of the multiplicative partition function, Talk at 2011 Illinois number theory meeting.

Paul Pollack, On the parity of the number of multiplicative partitions and related problems, Proc. Amer. Math. Soc. 140 (2012), 3793-3803.

A. Zaharescu and M. Zaki, On the parity of the number of multiplicative partitions, Acta Arith. 145 (2010), no. 3, 221-232. MR2733086. doi: 10.4064/aa145-3-2.

FORMULA

a(n) = A001055(n) mod 2.

PROG

(PARI) fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s}

vector(100, n, fcnt(n, n) % 2) \\ after Michael B. Porter in A001055, Michel Marcus, Jun 21 2015

CROSSREFS

Cf. A001055, A040051.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 24 2011

STATUS

approved

A050322

Number of factorizations indexed by prime signatures: A001055(A025487).

+20
9

1, 1, 2, 2, 3, 4, 5, 7, 5, 7, 9, 12, 11, 11, 16, 19, 21, 15, 29, 26, 30, 15, 31, 38, 22, 47, 52, 45, 36, 57, 64, 30, 77, 98, 67, 74, 97, 66, 105, 42, 109, 118, 92, 109, 171, 97, 141, 162, 137, 165, 56, 212, 181, 52, 198, 189, 289, 139, 250, 257, 269, 254, 77, 382, 267

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

For A025487(m) = 2^k = A000079(k), we have a(m) = A000041(k).

Is a(k) = A000110(k) for A025487(m) = A002110(k)?

LINKS

R. J. Mathar and Michael De Vlieger, Table of n, a(n) for n = 1..5000 (First 300 terms from R. J. Mathar)

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

EXAMPLE

From Gus Wiseman, Jan 13 2020: (Start)

The a(1) = 1 through a(11) = 9 factorizations:

{} 2 4 6 8 12 16 24 30 32 36

2*2 2*3 2*4 2*6 2*8 3*8 5*6 4*8 4*9

2*2*2 3*4 4*4 4*6 2*15 2*16 6*6

2*2*3 2*2*4 2*12 3*10 2*2*8 2*18

2*2*2*2 2*2*6 2*3*5 2*4*4 3*12

2*3*4 2*2*2*4 2*2*9

2*2*2*3 2*2*2*2*2 2*3*6

3*3*4

2*2*3*3

(End)

MAPLE

A050322 := proc(n)

A001055(A025487(n)) ;

end proc: # R. J. Mathar, May 25 2017

MATHEMATICA

c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* Michael De Vlieger, Jul 10 2017, after Dean Hickerson at A001055 *)

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Length/@facs/@First/@GatherBy[Range[1000], If[#==1, {}, Sort[Last/@FactorInteger[#]]]&] (* Gus Wiseman, Jan 13 2020 *)

CROSSREFS

Cf. A000041, A000079, A000110, A001055, A002110, A025487.

The version indexed by unsorted prime signature is A331049.

The version indexed by prime shadow (A181819, A181821) is A318284.

This sequence has range A045782 (same as A001055).

Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

KEYWORD

nonn

AUTHOR

Christian G. Bower, Oct 15 1999

STATUS

approved

A331023

Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).

+20
9

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 5, 1, 4, 1, 4, 1, 1, 1, 7, 2, 1, 3, 4, 1, 1, 1, 7, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 4, 4, 1, 1, 12, 2, 4, 1, 4, 1, 7, 1, 7, 1, 1, 1, 11, 1, 1, 4, 11, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 1, 12, 5, 1, 1, 11, 1, 1, 1, 7, 1, 11, 1, 4, 1, 1, 1, 19, 1, 4, 4, 9, 1, 1, 1, 7, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(2^n) = A330994(n).

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Table[Length[facs[n]]/Length[Select[facs[n], UnsameQ@@#&]], {n, 100}]//Numerator

PROG

(PARI)

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));

A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));

A331023(n) = numerator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

CROSSREFS

Positions of 1's are A005117.

Positions of 2's appear to be A001248.

The denominators are A331024.

The rounded quotients are A331048.

The same for integer partitions is A330994.

Cf. A001055, A001222, A002033, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jan 08 2020

EXTENSIONS

More terms from Antti Karttunen, May 27 2021

STATUS

approved

A331024

Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).

+20
9

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 7, 2, 1, 1, 9, 1, 1, 1, 5, 1, 9, 1, 3, 1, 1, 1, 10, 1, 3, 3, 5, 1, 1, 1, 5, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n

FORMULA

a(2^n) = A330995(n).

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Table[Length[facs[n]]/Length[Select[facs[n], UnsameQ@@#&]], {n, 100}]//Denominator

PROG

(PARI)

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));

A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));

A331024(n) = denominator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

CROSSREFS

Positions of 1's include all elements of A001248 as well as A005117. The first position of a 1 that is not in A167207 is 128.

The numerators are A331023.

The rounded quotients are A331048.

The same for integer partitions is A330995.

Cf. A001055, A005117, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

KEYWORD

nonn,frac

AUTHOR

Gus Wiseman, Jan 08 2020

EXTENSIONS

More terms from Antti Karttunen, May 27 2021

STATUS

approved

A331050

Positive integers whose number of factorizations into factors > 1 (A001055) is odd.

+20
9

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 100, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 125, 126, 127, 128

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

First differs from A319239 in lacking 256.

LINKS

Table of n, a(n) for n=1..63.

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

Select[Range[100], OddQ[Length[facs[#]]]&]

CROSSREFS

Complement of A331051.

The version for powers of two (instead of odds) is A330977.

The version for primes (instead of odds) is A330991.

Cf. A001055, A001221, A001222, A002033, A005117, A045778, A045782, A045783, A330972, A330973.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 10 2020

STATUS

approved

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