A037197 Numbers k such that tau(sigma(k)) = tau(k) where tau(k) is the number of divisors of k and sigma(k) their sum.

1, 2, 8, 12, 32, 52, 75, 84, 90, 98, 128, 150, 156, 338, 360, 392, 525, 528, 560, 600, 722, 867, 912, 972, 1050, 1352, 1452, 1456, 1525, 1734, 1922, 2064, 2160, 2340, 2400, 2888, 2890, 3050, 3120, 3216, 3698, 3744, 3872, 4080, 4144, 4200, 4500, 4575

Offset: 1

Author

Naohiro Nomoto

Keywords

• nonn

Examples

			k = 75: divisors(75) = {1, 3, 5, 15, 25, 75}, divisors(sigma(75)) = divisors(124) = {1, 2, 4, 31, 62, 124}, both 75 and sigma(75) have 6 divisors, so 75 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Crossrefs

Cf. A000005, A000203, A062068, A073802, A073803, A073804.

Programs

Mathematica

Do[s=DivisorSigma[0, DivisorSigma[1, n]]; s0=DivisorSigma[0, n]; If[Greater[s0, s], Print[n]], {n, 1, 1000}]
Select[Range[4600],DivisorSigma[0,#]==DivisorSigma[0,DivisorSigma[1,#]]&] (* Harvey P. Dale, Feb 08 2025 *)

PARI

is(n)=numdiv(sigma(n))==numdiv(n) \\ Charles R Greathouse IV, Feb 13 2013

Formula

Solutions to A000005(x) = A062068(x) = A000005(A000203(x)).

Conjecture: for n > 10^6, a(n) < n^2. - Benoit Cloitre, Aug 24 2002

Extensions

Offset corrected by Reinhard Zumkeller, Jun 18 2009

Name edited by Michel Marcus, Nov 12 2023