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

%I A277368 #18 Sep 08 2022 08:46:17
%S A277368 1,4,10,16,25,26,34,56,58,60,64,74,81,82,90,96,100,106,120,121,122,
%T A277368 132,146,178,184,194,202,204,216,218,226,234,248,274,276,289,298,306,
%U A277368 312,314,346,348,362,364,376,386,394,408,440,458,466,480,482,492,504,514
%N A277368 Numbers such that the number of their divisors divide the sum of their aliquot parts.
%C A277368 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
%D A277368 Richard G. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, chapter 2, p. 76.
%H A277368 Paolo P. Lava, <a href="/A277368/b277368.txt">Table of n, a(n) for n = 1..1000</a>
%H A277368 Paul T. Bateman, Paul Erdős, Carl Pomerance and E.G. Straus, <a href="https://doi.org/10.1007/BFb0096462">The arithmetic mean of the divisors of an integer</a>, 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, <a href="https://math.dartmouth.edu/~carlp/PDF/31.pdf">alternative link</a>.
%F A277368 Solutions k to A000005(k) | A001065(k).
%e A277368 sigma(26) - 26 = 42 - 26 = 16, d(26) = 4 and 16 / 4 = 4.
%p A277368 with(numtheory): P:= proc(q) local n; for n from 1 to q do
%p A277368 if type((sigma(n)-n)/tau(n),integer) then print(n); fi; od; end: P(10^3);
%t A277368 Select[Range@ 520, Mod[DivisorSigma[1, #] - #, DivisorSigma[0, #]] == 0 &] (* _Michael De Vlieger_, Oct 14 2016 *)
%o A277368 (PARI) isok(n) = ((sigma(n) - n) % numdiv(n)) == 0; \\ _Michel Marcus_, Oct 11 2016
%o A277368 (Magma) [k:k in [1..550]| (DivisorSigma(1,k)-k) mod DivisorSigma(0,k) eq 0]; // _Marius A. Burtea_, Jan 16 2020
%Y A277368 Cf. A000005, A001065, A003601, A047727.
%K A277368 nonn,easy
%O A277368 1,2
%A A277368 _Paolo P. Lava_, Oct 11 2016