A006039 Primitive nondeficient numbers. (Formerly M4132)

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

1,1

Comments

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

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.

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

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 (nondeficient), A005101 (abundant), A091191.

Subsequences: A000396 (perfect), A071395 (primitive abundant), A006038 (odd primitive abundant), A333967, A352739, A387709.

Positions of 1's in A341619 (characteristic function) and in A341620.

For equivalent sets of primitives for other abundancy ratios see A388019 (and its CROSSREFS).

Cf. A180332, A337688, A337689, A337690, A337691, A337814, A338133 (sorted by largest prime factor), A338427 (largest prime(n)-smooth), A342669.

Cf. also A337479.

Sequence in context: A324643 A119425 A342669 * A180332 A338133 A064771

Adjacent sequences: A006036 A006037 A006038 * A006040 A006041 A006042

Keyword

nonn

Author

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

Status

approved