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

1, 6, 12, 28, 30, 56, 117, 120, 132, 140, 182, 306, 380, 496, 552, 672, 775, 870, 992, 1080, 1287, 1406, 1428, 1680, 1722, 1892, 2016, 2184, 2256, 2480, 2793, 2862, 3276, 3540, 3640, 3782, 3960, 4060, 4556, 4560, 4650, 5112, 5382, 5402, 5460, 6120, 6320, 6552

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).

p is in: A053182 (k = 1), A065509 (k = 2), A163268 (k = 3), and A240693 (k = 5).

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.

Thus, A006516(n) and A000396(n), for n >= 1, are terms.

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

Cf. A000203, A000396, A006516, A036690, A053182, A064987, A065509, A163268, A327165, A327599.

Sequence in context: A032647 A327165 A338520 * A348034 A086792 A064987

Adjacent sequences: A339469 A339470 A339471 * A339473 A339474 A339475

Keyword

nonn

Author

Marius A. Burtea, Dec 06 2020

Status

approved