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

Author

Paolo P. Lava, Oct 11 2016

Keywords

• nonn
• easy

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

Examples

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

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.

Crossrefs

Cf. A000005, A001065, A003601, A047727.

Programs

Magma

[k:k in [1..550]| (DivisorSigma(1,k)-k) mod DivisorSigma(0,k) eq 0]; // Marius A. Burtea, Jan 16 2020

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 *)

PARI

isok(n) = ((sigma(n) - n) % numdiv(n)) == 0; \\ Michel Marcus, Oct 11 2016

Formula

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