A341620 -id:A341620

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A341624

a(n) = 0 if n is a deficient number, otherwise a(n) is the number of nondeficient divisors of the last number in the iteration x -> A003961(x) (starting from x=n) for which that count (A341620) is nonzero.

+20
4

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 6, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 1

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

OFFSET

1,12

LINKS

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

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

PROG

(PARI)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

A341620(n) = sumdiv(n, d, (sigma(d)>=(2*d)));

A341624(n) = { my(t, u=0); while((t=A341620(n))>0, u=t; n = A003961(n)); (u); };

CROSSREFS

Cf. A005100 (positions of zeros).

Differs from A341620 for the first time at n=120, where a(120)=1, while A341620(120)=8.

Cf. also A341508, A341618.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 22 2021

STATUS

approved

A006039

Primitive non-deficient numbers.
(Formerly M4132)

+10
20

6, 20, 28, 70, 88, 104, 272, 304, 368, 464, 496, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030

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

OFFSET

1,1

COMMENTS

A number n is non-deficient (A023196) iff it is abundant or perfect, that is iff A001065(n) is >= n. Since any multiple of a non-deficient number is itself non-deficient, we call a non-deficient number primitive iff all its proper divisors are deficient. - Jeppe Stig Nielsen, Nov 23 2003

Numbers whose proper multiples are all abundant, and whose proper divisors are all deficient. - Peter Munn, Sep 08 2020

As a set, shares with the sets of k-almost primes this property: no member divides another member and each positive integer not in the set is either a divisor of 1 or more members of the set or a multiple of 1 or more members of the set, but not both. See A337814 for proof etc. - Peter Munn, Apr 13 2022

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..8671

L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.

R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991

Jared Duker Lichtman, The reciprocal sum of primitive nondeficient numbers, Journal of Number Theory, Vol. 191 (2018), pp. 104-118.

Index entries for sequences where any odd perfect numbers must occur

FORMULA

Union of A000396 (perfect numbers) and A071395 (primitive abundant numbers). - M. F. Hasler, Jul 30 2016

Sum_{n>=1} 1/a(n) is in the interval (0.34842, 0.37937) (Lichtman, 2018). - Amiram Eldar, Jul 15 2020

MATHEMATICA

k = 1; lst = {}; While[k < 4050, If[DivisorSigma[1, k] >= 2 k && Min@Mod[k, lst] > 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Mar 09 2017 *)

CROSSREFS

Cf. A001065 (aliquot function), A023196 (non-deficient), A005101 (abundant), A091191.

Subsequences: A000396 (perfect), A071395 (primitive abundant), A006038 (odd primitive abundant), A333967, A352739.

Positions of 1's in A341620 and in A337690.

Cf. A180332, A337479, A337688, A337689, A337691, A337814, A338133 (sorted by largest prime factor), A338427 (largest prime(n)-smooth), A341619 (characteristic function), A342669.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, R. K. Guy

STATUS

approved

A294936

Characteristic function for nondeficient numbers (A023196): a(n) = 1 if A001065(n) >= n, 0 otherwise.

+10
12

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

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

OFFSET

COMMENTS

Differs from A210455 for the first time at n=70, and after that at n=836, 4030, 5830, 7192, 7912, 9272, etc., that is, at weird numbers, A006037.

Differs from A336835 for the first time at n=120. - Antti Karttunen, Apr 04 2022

LINKS

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

Index entries for characteristic functions

Index entries for sequences related to sums of divisors

FORMULA

a(n) = 1 if A033880(n) >= 0, 0 otherwise.

a(n) = 1 - A294934(n).

a(n) >= A210455(n).

a(n) >= A341619(n). - Antti Karttunen, Apr 04 2022

EXAMPLE

The proper divisors of 6 are 1, 2 and 3, and their sum is 6, and because 6 >= 6, a(6) = 1.

The proper divisors of 70 are 1, 2, 5, 7, 10, 14, and 35, and their sum is 74, and because 74 >= 70, a(70) = 1.

MATHEMATICA

Table[If[DivisorSigma[1, n]>=2n, 1, 0], {n, 120}] (* Harvey P. Dale, Mar 22 2020 *)

PROG

(PARI) a(n) = sigma(n, -1)>=2 \\ Felix Fröhlich, Nov 12 2017

CROSSREFS

Cf. A001065, A006037, A033880, A210455, A294887, A294934, A294935, A294937, A336835.

Cf. A023196 (positions of ones), A005100 (of zeros), A341620 (inverse Möbius transform), A294927 [= A341620(n)-a(n)], A341619.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 12 2017

STATUS

approved

A337690

a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

+10
11

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2

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

OFFSET

1,60

COMMENTS

As a simple consequence of the definition of a primitive nondeficient number, a(n) is nonzero if and only if n is nondeficient.

LINKS

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

Index entries for sequences related to sigma(n).

FORMULA

a(n) = Sum_{d|n} A341619(d) = Sum_{d|n} [1==A341620(d)]. - Corrected by Antti Karttunen, Feb 21 2021

a(A005100(n)) = 0.

a(A006039(n)) = 1.

a(A023196(n)) >= 1.

a(A337479(n)) = A337539(n).

a(n) <= A341620(n). - Antti Karttunen, Feb 22 2021

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A006039(n) = 0.3... (see A006039 for a better estimate of this constant). - Amiram Eldar, Jan 01 2024

EXAMPLE

The least nondeficient number, therefore the least primitive nondeficient number is 6. So a(1) = a(2) = a(3) = a(4) = a(5) = 0 as all primitive nondeficient numbers are larger, and therefore not divisors; and a(6) = 1, as only 1 primitive nondeficient number divides 6, namely 6 itself.

60 has the following 12 divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Of these, only 6 and 20 are in A006039, thus a(60) = 2.

PROG

(PARI)

A341619(n) = if(sigma(n) < (2*n), 0, fordiv(n, d, if((d<n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395

A337690(n) = sumdiv(n, d, A341619(d));

CROSSREFS

A006039 (or equivalently, its characteristic function, A341619) is used to define this sequence.

See A000203 and A023196 for definitions of deficient and nondeficient.

Sequences with similar definitions: A080224, A294927, A337539, A341620.

Cf. A302110, A341604.

Positions of 0's: A005100.

Positions of numbers >= k: A023196 (k=1), A337688 (k=2), A337689 (k=3).

Positions of first appearances are given in A337691.

Differs from its derived sequence A341618 for the first time at n=120, where a(120)=2, while A341618(120)=1.

KEYWORD

nonn

AUTHOR

Antti Karttunen and Peter Munn, Sep 15 2020

EXTENSIONS

Data section extended to 120 terms by Antti Karttunen, Feb 21 2021

STATUS

approved

A341619

Characteristic function of primitive nondeficient numbers (A006039): a(n) = 1 if proper multiples of n are all abundant, and proper divisors of n are all deficient, 0 otherwise.

+10
7

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

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

OFFSET

LINKS

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

Index entries for characteristic functions

Index entries for sequences related to sigma(n)

FORMULA

a(n) = [A341620(n) == 1], where [ ] is the Iverson bracket.

a(n) = 1 iff A341620(n) = 1.

MATHEMATICA

a[n_] := Boole[DivisorSum[n, 1 &, DivisorSigma[1, #] >= 2*# &] == 1]; Array[a, 100] (* Amiram Eldar, Feb 22 2021 *)

PROG

(PARI) A341619(n) = if(sigma(n)<(2*n), 0, fordiv(n, d, if((d<n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395

CROSSREFS

Cf. A006039, A071395, A337690 (inverse Möbius transform), A341620.

Cf. also A341609.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 21 2021

STATUS

approved

A357460

Numbers whose number of deficient divisors is equal to their number of nondeficient divisors.

+10
3

72, 108, 120, 168, 180, 252, 420, 528, 560, 624, 1188, 1224, 1368, 1400, 1404, 1632, 1656, 1824, 1836, 1960, 1980, 2040, 2052, 2088, 2208, 2232, 2280, 2340, 2484, 2664, 2760, 2772, 2784, 2856, 2952, 2976, 3060, 3096, 3132, 3192, 3200, 3276, 3348, 3384, 3420, 3432

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

OFFSET

1,1

COMMENTS

Numbers k such that A080226(k) = A341620(k).

This sequence is infinite: if p >= 17 is a prime then 72*p is a term.

The least odd term of this sequence is a(36126824) = A357461(1) = 3010132125.

Since the number of divisors of any term is even, none of the terms are squares.

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 1, 10, 131, 1172, 12003, 120647, 1199147, 11992293, 120089446, ... . Apparently, the asymptotic density of this sequence exists and is equal to about 0.012.

LINKS

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

EXAMPLE

72 is a term since it has 12 divisors, 6 of them (1, 2, 3, 4, 8 and 9) are deficient and 6 (6, 12, 18, 24, 36 and 72) are not.

MATHEMATICA

q[n_] := DivisorSum[n, If[DivisorSigma[-1, #] < 2, 1, -1] &] == 0; Select[Range[3500], q]

PROG

(PARI) is(n) = sumdiv(n, d, if(sigma(d, -1) < 2, 1, -1)) == 0;

CROSSREFS

Subsequence of A000037 and A005101.

Cf. A080226, A335543, A335544, A341620, A357461, A357462.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Sep 29 2022

STATUS

approved

A342669

Even numbers which are either primitively nondeficient (A006039), or become such after applying prime shift A003961 some number of times to them.

+10
2

6, 20, 28, 70, 88, 104, 120, 180, 272, 300, 304, 368, 420, 464, 496, 504, 550, 572, 630, 650, 660, 748, 780, 836, 924, 990, 1020, 1050, 1092, 1140, 1170, 1184, 1312, 1376, 1380, 1430, 1470, 1504, 1650, 1696, 1740, 1860, 1870, 1888, 1952, 2002, 2090, 2210, 2220, 2310, 2460, 2470, 2530, 2580, 2584, 2730, 2820, 2856, 2990

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

OFFSET

1,1

COMMENTS

Even numbers k for which A341624(k) = 1.

Even numbers whose closure under map x -> A003961(x) contains a primitive non-deficient number (one of the terms of A006039). Shifting each term k exactly A336835(k)-1 times with A003961 towards larger primes gives those numbers, but not in monotonic order, producing instead a permutation of A006039.

Sequence 2*A246277(A006039(.)), sorted into ascending order.

If there are any two terms, x and y, such that the other is a multiple of the other, then A336835(x) != A336835(y), and furthermore, for any term k present here, for all its proper divisors (d|k, d<k) it holds that A336835(d) < A336835(k), in other words, they reach the deficiency earlier (by prime shifting) than k itself.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sigma(n)

Index entries for sequences computed from indices in prime factorization

EXAMPLE

For n = 120 = 2^3 * 3 * 5, A341620(120) = 8, so it is not primitive nondeficient. However, prime-shifting it once gives A003961(120) = 945 = 3^3 * 5 * 7, which is one of the terms of A006039 as A341620(945) = 1. Therefore 120 is included in the sequence.

PROG

(PARI)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

A341620(n) = sumdiv(n, d, (sigma(d)>=(2*d)));

A341624(n) = { my(t, u=0); while((t=A341620(n))>0, u=t; n = A003961(n)); (u); };

isA342669(n) = (!(n%2)&&(1==A341624(n)));

CROSSREFS

Cf. A000396, A006039 (even terms of these form a subsequence).

Cf. A003961, A246277, A336835, A341605/A341606, A341620, A341624.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 20 2021

STATUS

approved

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