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A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).

%I A339472 #40 Dec 22 2024 14:28:04
%S A339472 1,6,12,28,30,56,117,120,132,140,182,306,380,496,552,672,775,870,992,
%T A339472 1080,1287,1406,1428,1680,1722,1892,2016,2184,2256,2480,2793,2862,
%U A339472 3276,3540,3640,3782,3960,4060,4556,4560,4650,5112,5382,5402,5460,6120,6320,6552
%N A339472 Integers k for which there is a divisor d, such that sigma(k) = d*sigma(d).
%C A339472 All terms are nonprimes.
%C A339472 The sequence includes all numbers of the form p*(p + 1) with p prime. Indeed: sigma(p*(p + 1)) = sigma(p)*sigma(p + 1) = (p + 1)*sigma(p + 1). So A036690 is a subsequence. Thus, the sequence is infinite.
%C A339472 Let k >= 1. If p and q = 1 + p + ... + p^(2*k) are prime numbers, then m = p^(2*k)*q is a term. Indeed, sigma(m) = sigma(p^(2*k)*q) = sigma(p^(2*k))*sigma(q) = q*sigma(q).
%C A339472 p is in: A053182 (k = 1), A065509 (k = 2), A163268 (k = 3), and A240693 (k = 5).
%C A339472 For k = 4 there are no prime p because 1 + p + p^2 + p^3 + p^4 + p^5 + p^6 + p^7 + p^8 = (p^6 + p^3 + 1)*(p^2 + p + 1).
%C A339472 If m = 2^(p - 1)*(2^p - 1), p >= 1, (see A006516), then sigma(m) = sigma(2^(p - 1)*(2^p - 1)) = sigma(2^(p - 1))*sigma(2^p - 1) = (2^p - 1)*sigma(2^p - 1), so m is a term.
%C A339472 Thus, A006516(n) and A000396(n), for n >= 1, are terms.
%H A339472 Seiichi Manyama, <a href="/A339472/b339472.txt">Table of n, a(n) for n = 1..5000</a>
%e A339472 sigma(6) = 12 = 3*4 = 3*sigma(3), so 6 is a term.
%e A339472 sigma(12) = 28 = 4*7 = 4*sigma(4), so 12 is a term.
%e A339472 sigma(30) = 72 = 6*12 = 6*sigma(6), so 30 is a term.
%e A339472 sigma(56) = 120 = 8*15 = 8*sigma(8), so 56 is a term.
%e A339472 sigma(117) = 182 = 13*14 = 13*sigma(13), so 117 is a term.
%t A339472 q[n_] := Module[{d = Divisors[n], s}, s = Plus @@ d; AnyTrue[d, #*DivisorSigma[1, #] == s &]]; Select[Range[7000], q] (* _Amiram Eldar_, Dec 06 2020 *)
%o A339472 (Magma) s:=func<n|exists(u){d:d in Divisors(n)|DivisorSigma(1,n) eq DivisorSigma(1,d)*d}>; [n:n in [1..6600]|s(n)];
%o A339472 (PARI) isok(k) = my(sk=sigma(k)); fordiv(k, d, if (d*sigma(d) == sk, return(1))); \\ _Michel Marcus_, Dec 06 2020
%Y A339472 Cf. A000203, A000396, A006516, A036690, A053182, A064987, A065509, A163268, A327165, A327599.
%K A339472 nonn
%O A339472 1,2
%A A339472 _Marius A. Burtea_, Dec 06 2020