A277368 Numbers such that the number of their divisors divide the sum of their aliquot parts.

1, 4, 10, 16, 25, 26, 34, 56, 58, 60, 64, 74, 81, 82, 90, 96, 100, 106, 120, 121, 122, 132, 146, 178, 184, 194, 202, 204, 216, 218, 226, 234, 248, 274, 276, 289, 298, 306, 312, 314, 346, 348, 362, 364, 376, 386, 394, 408, 440, 458, 466, 480, 482, 492, 504, 514

Offset

1,2

Comments

If p is a prime such that p == 2 (mod 3) then p^2 is a term. Bateman et al. (1981) proved that the asymptotic density of this sequence is 0. - Amiram Eldar, Jan 16 2020

References

Richard G. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, chapter 2, p. 76.

Links

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Paul T. Bateman, Paul Erdős, Carl Pomerance and E.G. Straus, The arithmetic mean of the divisors of an integer, in Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol. 899, Springer, Berlin - New York, 1981, pp. 197-220, alternative link.

Formula

Solutions k to A000005(k) | A001065(k).

Example

sigma(26) - 26 = 42 - 26 = 16, d(26) = 4 and 16 / 4 = 4.

Maple

with(numtheory): P:= proc(q) local n; for n from 1 to q do
if type((sigma(n)-n)/tau(n), integer) then print(n); fi; od; end: P(10^3);

Mathematica

Select[Range@ 520, Mod[DivisorSigma[1, #] - #, DivisorSigma[0, #]] == 0 &] (* Michael De Vlieger, Oct 14 2016 *)

Prog

(PARI) isok(n) = ((sigma(n) - n) % numdiv(n)) == 0; \\ Michel Marcus, Oct 11 2016
(Magma) [k:k in [1..550]| (DivisorSigma(1, k)-k) mod DivisorSigma(0, k) eq 0]; // Marius A. Burtea, Jan 16 2020

Crossrefs

Cf. A000005, A001065, A003601, A047727.

Sequence in context: A366661 A271911 A322948 * A067274 A331081 A054901

Adjacent sequences: A277365 A277366 A277367 * A277369 A277370 A277371

Keyword

nonn,easy

Author

Paolo P. Lava, Oct 11 2016

Status

approved