OFFSET
1,2
COMMENTS
All terms are nonprimes.
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.
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).
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).
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.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..5000
EXAMPLE
sigma(6) = 12 = 3*4 = 3*sigma(3), so 6 is a term.
sigma(12) = 28 = 4*7 = 4*sigma(4), so 12 is a term.
sigma(30) = 72 = 6*12 = 6*sigma(6), so 30 is a term.
sigma(56) = 120 = 8*15 = 8*sigma(8), so 56 is a term.
sigma(117) = 182 = 13*14 = 13*sigma(13), so 117 is a term.
MATHEMATICA
q[n_] := Module[{d = Divisors[n], s}, s = Plus @@ d; AnyTrue[d, #*DivisorSigma[1, #] == s &]]; Select[Range[7000], q] (* Amiram Eldar, Dec 06 2020 *)
PROG
(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)];
(PARI) isok(k) = my(sk=sigma(k)); fordiv(k, d, if (d*sigma(d) == sk, return(1))); \\ Michel Marcus, Dec 06 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Dec 06 2020
STATUS
approved