A337688 -id:A337688 - cp's OEIS frontend

A006039 Primitive nondeficient numbers.

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

Offset: 1

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

nonn

A number n is nondeficient (A023196) iff it is abundant or perfect, that is iff A001065(n) is >= n. Since any multiple of a nondeficient number is itself nondeficient, we call a nondeficient 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

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

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.
Joshua Zelinsky, The Sum of the Reciprocals of the Prime Divisors of an Odd Perfect or Odd Primitive Non-deficient Number, Integers (2025) Vol. 25, Art. No. A59. See p. 2.
Index entries for sequences where any odd perfect numbers must occur

Cf. A001065 (aliquot function), A023196 (nondeficient), 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.

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 *)

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

A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Peter Munn, Sep 15 2020

nonn

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

			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.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n).

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.

A341619(n) = if(sigma(n) < (2*n), 0, fordiv(n, d, if((d= 2*d), return(0))); (1)); \\ After code in A071395
A337690(n) = sumdiv(n, d, A341619(d));

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

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

A337689 Numbers divisible by 3 or more primitive nondeficient numbers.

Original entry on oeis.org

140, 280, 420, 560, 700, 840, 980, 1120, 1144, 1260, 1320, 1400, 1540, 1560, 1680, 1820, 1848, 1890, 1960, 2100, 2184, 2200, 2240, 2288, 2380, 2520, 2600, 2640, 2660, 2800, 2860, 2940, 2992, 3080, 3120, 3150, 3220, 3300, 3344, 3360, 3432, 3500, 3640, 3696, 3740

Offset: 1

David A. Corneth, Antti Karttunen and Peter Munn, Sep 20 2020

A006039 lists the primitive nondeficient numbers.

			Table of n, a(n) and the relevant divisors starts:
   n   a(n)   divisors in A006039
   1    140   20, 28, 70;
   2    280   20, 28, 70;
   3    420   6, 20, 28, 70;
   4    560   20, 28, 70;
   5    700   20, 28, 70;
   6    840   6, 20, 28, 70;
   7    980   20, 28, 70;
   8   1120   20, 28, 70;
   9   1144   88, 104, 572;
  10   1260   6, 20, 28, 70;
  11   1320   6, 20, 88;
  12   1400   20, 28, 70;

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

A006039, A337690 are used in a definition of this sequence.

Subsequence of: A337688.

See Corneth link \\ David A. Corneth, Oct 07 2020

Numbers k such that A337690(k) >= 3.

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A006039 Primitive nondeficient numbers.

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A337690 a(n) is the number of primitive nondeficient numbers (A006039) dividing n.

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A337689 Numbers divisible by 3 or more primitive nondeficient numbers.

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