A181819 -id:A181819

A182850

a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.

+20
126

0, 0, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 2, 3, 3, 3, 3, 1, 3, 3, 4, 1, 3, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 3, 1, 4, 3, 3, 1, 4, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 4, 1, 4, 4, 3, 1, 3, 1, 4, 3

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

OFFSET

1,4

COMMENTS

The fixed points of the x -> A181819(x) map are 1 and 2. Note that the x -> A000005(x) map has the same fixed points, and that A000005(n) = A181819(n) iff n is cubefree (cf. A004709). Under the x -> A181819(x) map, it seems significantly easier to generalize about which kinds of integers take a given number of iterations to reach a fixed point than under the x -> A000005(x) map.

Also the number of steps in the reduction of the multiset of prime factors of n wherein one repeatedly takes the multiset of multiplicities. For example, the a(90) = 5 steps are {2,3,3,5} -> {1,1,2} -> {1,2} -> {1,1} -> {2} -> {1}. - Gus Wiseman, May 13 2018

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Fixed Point

Eric Weisstein's World of Mathematics, Map

FORMULA

For n > 2, a(n) = a(A181819(n)) + 1.

a(n) = 0 iff n equals 1 or 2.

a(n) = 1 iff n is an odd prime (A000040(n) for n>1).

a(n) = 2 iff n is a composite perfect prime power (A025475(n) for n>1).

a(n) = 3 iff n is a squarefree composite integer or a power of a squarefree composite integer (cf. A182853).

a(n) = 4 iff n's prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number (cf. A182854).

EXAMPLE

A181819(6) = 4; A181819(4) = 3; A181819(3) = 2; A181819(2) = 2. Therefore, a(6) = 3, a(4) = 2, a(3)= 1, and a(2) = 0.

MATHEMATICA

Table[If[n<=2, 0, Length[FixedPointList[Sort[Length/@Split[#]]&, Sort[Last/@FactorInteger[n]]]]-1], {n, 100}] (* Gus Wiseman, May 13 2018 *)

PROG

(Haskell)

a182850 n = length $ takeWhile (`notElem` [1, 2]) $ iterate a181819 n

-- Reinhard Zumkeller, Mar 26 2012

(Scheme, with memoization-macro definec)

(definec (A182850 n) (if (<= n 2) 0 (+ 1 (A182850 (A181819 n))))) ;; Antti Karttunen, Feb 05 2016

CROSSREFS

A182857 gives values of n where a(n) increases to a record.

Cf. A000961, A001222, A003434, A005117, A007916, A036459, A112798, A130091, A181819, A182851-A182858, A238748, A304455, A304464, A304465.

KEYWORD

nonn

AUTHOR

Matthew Vandermast, Jan 04 2011

STATUS

approved

A323014

a(1) = 0; a(prime) = 1; otherwise a(n) = 1 + a(A181819(n)).

+20
93

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 2, 3, 3, 3, 3, 1, 3, 3, 4, 1, 3, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 3, 1, 4, 3, 3, 1, 4, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 4, 1, 4, 4, 3, 1, 3, 1, 4, 3

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

OFFSET

1,4

COMMENTS

Except for n = 2, same as A182850. Unlike A182850, the terms of this sequence depend only on the prime signature (A101296, A118914) of the index.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

FORMULA

For all n >= 1, a(n) = a(A046523(n)). [See comment] - Antti Karttunen, Jun 10 2022

MATHEMATICA

dep[n_]:=If[n==1, 0, If[PrimeQ[n], 1, 1+dep[Times@@Prime/@Last/@FactorInteger[n]]]];

Array[dep, 100]

PROG

(PARI)

A181819(n) = factorback(apply(e->prime(e), (factor(n)[, 2])));

A323014(n) = if(1==n, 0, if(isprime(n), 1, 1+A323014(A181819(n)))); \\ Antti Karttunen, Jun 10 2022

CROSSREFS

Positions of 1's are the prime numbers A000040.

Positions of 2's are the proper prime powers A246547.

Positions of 3's are A182853.

Row lengths of A323023.

Cf. A001221, A001222, A046523, A056239, A070175, A071625, A101296, A112798, A118914, A181819, A181821, A182850, A182857, A304465, A323022.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 02 2019

EXTENSIONS

Terms a(88) and beyond from Antti Karttunen, Jun 10 2022

STATUS

approved

A182857

Smallest number that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

+20
56

1, 3, 4, 6, 12, 60, 2520, 1286485200, 35933692027611398678865941374040400000

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

OFFSET

0,2

COMMENTS

a(9) has 296 digits.

Related to Levine's sequence (A011784): A011784(n) = A001222(a(n)) = A001221(a(n+1)) = A051903(a(n+2)) = A071625(a(n+2)). Also see A182858.

Values of n where A182850(n) increases to a record.

The multiplicity of prime(k) in a(n+1) is the k-th largest prime index of a(n), which is A296150(a(n),k). - Gus Wiseman, May 13 2018

LINKS

Gus Wiseman, Table of n, a(n) for n = 0..9

FORMULA

For n > 0, a(n) = A181819(a(n+1)). For n > 1, a(n) = A181821(a(n-1)).

EXAMPLE

From Gus Wiseman, May 13 2018: (Start)

Like A001462 the following sequence of multisets whose Heinz numbers belong to this sequence is a run-length describing sequence, as the number of k's in row n + 1 is equal to the k-th term of row n.

{2}

{1,1}

{1,2}

{1,1,2}

{1,1,2,3}

{1,1,1,2,2,3,4}

{1,1,1,1,2,2,2,3,3,4,4,5,6,7}

{1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7,7,7,8,8,9,9,10,10,11,12,13,14}

(End)

MATHEMATICA

Prepend[Function[m, Times@@Prime/@m]/@NestList[Join@@Table[Table[i, {Reverse[#][[i]]}], {i, Length[#]}]&, {2}, 8], 1] (* Gus Wiseman, May 13 2018 *)

CROSSREFS

Cf. A001462, A007755, A007916, A009287, A012257, A112798, A181819, A182850-A182858, A296150, A304455, A304464, A304465.

KEYWORD

nonn

AUTHOR

Matthew Vandermast, Jan 05 2011

STATUS

approved

A325277

Irregular triangle read by rows where row 1 is {1} and row n is the sequence starting with n and repeatedly applying A181819 until a prime number is reached.

+20
48

1, 2, 3, 4, 3, 5, 6, 4, 3, 7, 8, 5, 9, 3, 10, 4, 3, 11, 12, 6, 4, 3, 13, 14, 4, 3, 15, 4, 3, 16, 7, 17, 18, 6, 4, 3, 19, 20, 6, 4, 3, 21, 4, 3, 22, 4, 3, 23, 24, 10, 4, 3, 25, 3, 26, 4, 3, 27, 5, 28, 6, 4, 3, 29, 30, 8, 5, 31, 32, 11, 33, 4, 3

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

OFFSET

1,2

COMMENTS

The function A181819 maps p^i*...*q^j to prime(i)*...*prime(j) where p through q are distinct primes.

LINKS

Table of n, a(n) for n=1..73.

FORMULA

T(n,k) = A325239(n,k) for k <= A323014(n).

A001222(T(n,k)) = A323023(n,k) for n > 1.

EXAMPLE

Triangle begins:

1 26 4 3 51 4 3 76 6 4 3

2 27 5 52 6 4 3 77 4 3

3 28 6 4 3 53 78 8 5

4 3 29 54 10 4 3 79

5 30 8 5 55 4 3 80 14 4 3

6 4 3 31 56 10 4 3 81 7

7 32 11 57 4 3 82 4 3

8 5 33 4 3 58 4 3 83

9 3 34 4 3 59 84 12 6 4 3

10 4 3 35 4 3 60 12 6 4 3 85 4 3

11 36 9 3 61 86 4 3

12 6 4 3 37 62 4 3 87 4 3

13 38 4 3 63 6 4 3 88 10 4 3

14 4 3 39 4 3 64 13 89

15 4 3 40 10 4 3 65 4 3 90 12 6 4 3

16 7 41 66 8 5 91 4 3

17 42 8 5 67 92 6 4 3

18 6 4 3 43 68 6 4 3 93 4 3

19 44 6 4 3 69 4 3 94 4 3

20 6 4 3 45 6 4 3 70 8 5 95 4 3

21 4 3 46 4 3 71 96 22 4 3

22 4 3 47 72 15 4 3 97

23 48 14 4 3 73 98 6 4 3

24 10 4 3 49 3 74 4 3 99 6 4 3

25 3 50 6 4 3 75 6 4 3 100 9 3

MATHEMATICA

red[n_]:=Times@@Prime/@Last/@If[n==1, {}, FactorInteger[n]];

Table[NestWhileList[red, n, #>1&&!PrimeQ[#]&], {n, 30}]

CROSSREFS

Row lengths are 1 for n = 1 and A323014(n) for n > 1.

Cf. A001221, A001222, A071625, A118914, A181819, A181821, A182850, A182857, A323022, A323023, A325238, A325239.

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Apr 15 2019

STATUS

approved

A325755

Numbers n divisible by their prime shadow A181819(n).

+20
32

1, 2, 9, 12, 18, 36, 40, 60, 84, 112, 120, 125, 132, 156, 180, 204, 225, 228, 250, 252, 276, 280, 336, 348, 352, 360, 372, 396, 440, 441, 444, 450, 468, 492, 516, 520, 540, 560, 564, 600, 612, 636, 675, 680, 684, 708, 732, 760, 804, 828, 832, 840, 852, 876

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

OFFSET

1,2

COMMENTS

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions containing their multiset of multiplicities as a submultiset (counted by A325702).

LINKS

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

EXAMPLE

The sequence of terms together with their prime indices begins:

1: {}

2: {1}

9: {2,2}

12: {1,1,2}

18: {1,2,2}

36: {1,1,2,2}

40: {1,1,1,3}

60: {1,1,2,3}

84: {1,1,2,4}

112: {1,1,1,1,4}

120: {1,1,1,2,3}

125: {3,3,3}

132: {1,1,2,5}

156: {1,1,2,6}

180: {1,1,2,2,3}

204: {1,1,2,7}

225: {2,2,3,3}

228: {1,1,2,8}

250: {1,3,3,3}

252: {1,1,2,2,4}

MATHEMATICA

red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];

Select[Range[100], Divisible[#, red[#]]&]

CROSSREFS

Cf. A181819, A182857, A290689, A290822, A323014, A324843, A325702, A325706, A325708, A325756.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 19 2019

STATUS

approved

A304660

A run-length describing inverse to A181819. The multiplicity of prime(k) in a(n) is the k-th smallest prime index of n, which is A112798(n,k).

+20
30

1, 2, 4, 6, 8, 18, 16, 30, 36, 54, 32, 150, 64, 162, 108, 210, 128, 450, 256, 750, 324, 486, 512, 1470, 216, 1458, 900, 3750, 1024, 2250, 2048, 2310, 972, 4374, 648, 7350, 4096, 13122, 2916, 10290, 8192, 11250, 16384, 18750, 4500, 39366, 32768, 25410, 1296

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

OFFSET

1,2

COMMENTS

A permutation of A133808. a(n) is the smallest member m of A133808 such that A181819(m) = n.

LINKS

Table of n, a(n) for n=1..49.

FORMULA

a(n) = Product_{i = 1..Omega(n)} prime(i)^A112798(n,i).

EXAMPLE

Sequence of normalized prime multisets together with the normalized prime multisets of their images begins:

1: {} -> {}

2: {1} -> {1}

3: {2} -> {1,1}

4: {1,1} -> {1,2}

5: {3} -> {1,1,1}

6: {1,2} -> {1,2,2}

7: {4} -> {1,1,1,1}

8: {1,1,1} -> {1,2,3}

9: {2,2} -> {1,1,2,2}

10: {1,3} -> {1,2,2,2}

11: {5} -> {1,1,1,1,1}

12: {1,1,2} -> {1,2,3,3}

13: {6} -> {1,1,1,1,1,1}

14: {1,4} -> {1,2,2,2,2}

15: {2,3} -> {1,1,2,2,2}

16: {1,1,1,1} -> {1,2,3,4}

17: {7} -> {1,1,1,1,1,1,1}

18: {1,2,2} -> {1,2,2,3,3}

MATHEMATICA

Table[With[{y=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]}, Times@@Power[Array[Prime, Length[y]], y]], {n, 100}]

CROSSREFS

Cf. A055932, A056239, A112798, A130091, A133808, A181819, A181821, A182850, A182857, A275870, A304455.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 16 2018

STATUS

approved

A325239

Irregular triangle read by rows where row 1 is {1} and row n > 1 is the sequence starting with n and repeatedly applying A181819 until 2 is reached.

+20
17

1, 2, 3, 2, 4, 3, 2, 5, 2, 6, 4, 3, 2, 7, 2, 8, 5, 2, 9, 3, 2, 10, 4, 3, 2, 11, 2, 12, 6, 4, 3, 2, 13, 2, 14, 4, 3, 2, 15, 4, 3, 2, 16, 7, 2, 17, 2, 18, 6, 4, 3, 2, 19, 2, 20, 6, 4, 3, 2, 21, 4, 3, 2, 22, 4, 3, 2, 23, 2, 24, 10, 4, 3, 2, 25, 3, 2, 26, 4, 3, 2

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

OFFSET

1,2

COMMENTS

The function A181819 maps n = p^i*...*q^j to prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n.

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10581 (rows 1..2500, flattened)

Michael De Vlieger, For m in row n, plot black else plot white, n = 1..120, 4X magnification.

Michael De Vlieger, For m in row n, plot black else plot white, n = 1..2^12.

FORMULA

A001222(T(n,k)) = A323023(n,k), n > 2, k <= A182850(n).

EXAMPLE

Triangle begins:

1 26 4 3 2 51 4 3 2 76 6 4 3 2

2 27 5 2 52 6 4 3 2 77 4 3 2

3 2 28 6 4 3 2 53 2 78 8 5 2

4 3 2 29 2 54 10 4 3 2 79 2

5 2 30 8 5 2 55 4 3 2 80 14 4 3 2

6 4 3 2 31 2 56 10 4 3 2 81 7 2

7 2 32 11 2 57 4 3 2 82 4 3 2

8 5 2 33 4 3 2 58 4 3 2 83 2

9 3 2 34 4 3 2 59 2 84 12 6 4 3 2

10 4 3 2 35 4 3 2 60 12 6 4 3 2 85 4 3 2

11 2 36 9 3 2 61 2 86 4 3 2

12 6 4 3 2 37 2 62 4 3 2 87 4 3 2

13 2 38 4 3 2 63 6 4 3 2 88 10 4 3 2

14 4 3 2 39 4 3 2 64 13 2 89 2

15 4 3 2 40 10 4 3 2 65 4 3 2 90 12 6 4 3 2

16 7 2 41 2 66 8 5 2 91 4 3 2

17 2 42 8 5 2 67 2 92 6 4 3 2

18 6 4 3 2 43 2 68 6 4 3 2 93 4 3 2

19 2 44 6 4 3 2 69 4 3 2 94 4 3 2

20 6 4 3 2 45 6 4 3 2 70 8 5 2 95 4 3 2

21 4 3 2 46 4 3 2 71 2 96 22 4 3 2

22 4 3 2 47 2 72 15 4 3 2 97 2

23 2 48 14 4 3 2 73 2 98 6 4 3 2

24 10 4 3 2 49 3 2 74 4 3 2 99 6 4 3 2

25 3 2 50 6 4 3 2 75 6 4 3 2 100 9 3 2

MATHEMATICA

red[n_]:=Times@@Prime/@Last/@If[n==1, {}, FactorInteger[n]];

Table[NestWhileList[red, n, #>2&], {n, 30}]

CROSSREFS

Row lengths are A182850(n) + 1.

Cf. A001221, A001222, A071625, A118914, A181819, A181821, A182857, A323014, A323022, A323023, A325238, A325277.

See A353510 for a full square array version of this table.

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Apr 15 2019

STATUS

approved

A353393

Positive integers m > 1 that are prime or whose prime shadow A181819(m) is a divisor of m that is already in the sequence.

+20
13

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 225, 227, 229, 233, 239, 241, 251

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

OFFSET

1,1

COMMENTS

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

LINKS

Table of n, a(n) for n=1..58.

FORMULA

Equals A353389 U A000040.

EXAMPLE

The terms together with their prime indices begin:

2: {1}

3: {2}

5: {3}

7: {4}

9: {2,2}

11: {5}

13: {6}

17: {7}

19: {8}

23: {9}

29: {10}

31: {11}

36: {1,1,2,2}

MATHEMATICA

red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];

suQ[n_]:=PrimeQ[n]||Divisible[n, red[n]]&&suQ[red[n]];

Select[Range[2, 200], suQ[#]&]

CROSSREFS

The first term that is not a prime power A000961 is 36.

The first term that is not a perfect power A001597 is 1260.

The non-recursive version is A325755, counted by A325702.

Removing all primes gives A353389.

These partitions are counted by A353426.

The version for compositions is A353431.

A001222 counts prime factors with multiplicity, distinct A001221.

A003963 gives product of prime indices.

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914.

A130091 lists numbers with all distinct prime exponents, counted by A098859.

A181819 gives prime shadow, with an inverse A181821.

A325131 lists numbers relatively prime to their prime shadow.

Cf. A000005, A000040, A047966, A182850, A316413, A316428, A325756, A353394, A353395, A353399.

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 15 2022

STATUS

approved

A305732

Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.

+20
12

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

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

OFFSET

1,1

COMMENTS

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n. A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1,...,m_k) = 1 and the multiset {y_1,...,y_k} is also reducible.

LINKS

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

EXAMPLE

60 has relatively prime prime indices {1,1,2,3} with multiplicities {1,1,2} corresponding to A181819(90) = 12. 12 has relatively prime prime indices {1,1,2} with multiplicities {1,2} corresponding to A181819(12) = 6. 6 has relatively prime prime indices {1,2} with multiplicities {1,1} corresponding to A181819(6) = 4. 4 has relatively prime prime indices {1,1} with multiplicities {2} corresponding to A181819(4) = 3. 3 is prime, so we conclude that 60 belongs to the sequence.

MATHEMATICA

rdzQ[n_]:=And[n>1, Or[PrimeQ[n], And[rdzQ[Times@@Prime/@FactorInteger[n][[All, 2]]], GCD@@PrimePi/@FactorInteger[n][[All, 1]]==1]]];

Select[Range[50], rdzQ]

CROSSREFS

Cf. A000837, A007916, A056239, A181819, A182850, A289508, A289509, A298748, A304465, A304687, A304818, A305563, A305731, A305733.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 22 2018

STATUS

approved

A182855

Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

+20
9

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

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

OFFSET

1,1

COMMENTS

In each case, 2 is the fixed point that is reached (1 is the other fixed point of the x -> A181819(x) map).

Includes all integers whose prime signature a) contains two or more distinct numbers, and b) contains no number that occurs the same number of times as any other number. The first member of this sequence that does not fit that description is 75675600, whose prime signature is (4,3,2,2,1,1).

A full characterization is: Numbers whose prime signature (1) has not all equal multiplicities but (2) the numbers of distinct parts appearing with each distinct multiplicity are all equal. For example, the prime signature of 2520 is {1,1,2,3}, which satisfies (1) but fails (2), as the numbers of distinct parts appearing with each distinct multiplicity are 1 (with multiplicity 2, the part being 1) and 2 (with multiplicity 1, the parts being 2 and 3). Hence the sequence does not contain 2520. - Gus Wiseman, Jan 02 2019

LINKS

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

Eric Weisstein's World of Mathematics, Fixed Point

Eric Weisstein's World of Mathematics, Map

EXAMPLE

1. 180 requires exactly five iterations under the x -> A181819(x) map to reach a fixed point (namely, 2). A181819(180) = 18; A181819(18) = 6; A181819(6) = 4; A181819(4) = 3; A181819(3) = 2 (and A181819(2) = 2).

2. The prime signature of 180 (2^2*3^2*5) is (2,2,1).

a. Two distinct numbers appear in (2,2,1) (namely, 1 and 2).

b. Neither 1 nor 2 appears in (2,2,1) the same number of times as any other number that appears there.

MATHEMATICA

Select[Range[1000], With[{sig=Sort[Last/@FactorInteger[#]]}, And[!SameQ@@Length/@Split[sig], SameQ@@Length/@Union/@GatherBy[sig, Length[Position[sig, #]]&]]]&] (* Gus Wiseman, Jan 02 2019 *)

CROSSREFS

Numbers n such that A182850(n) = 5. See also A182853, A182854.

Subsequence of A059404 and A182851. Includes A085987 and A179642 as subsequences.

Cf. A001221, A001222, A006939, A062770, A071625, A118914, A181819, A181821, A182857, A323014, A323022, A323023.

KEYWORD

nonn

AUTHOR

Matthew Vandermast, Jan 04 2011

STATUS

approved