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

1,2

Links

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

Formula

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

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A126775 A306898 A069185 * A125712 A062290 A176961

Adjacent sequences: A037194 A037195 A037196 * A037198 A037199 A037200

Keyword

nonn,changed

Author

Naohiro Nomoto

Extensions

Offset corrected by Reinhard Zumkeller, Jun 18 2009

Name edited by Michel Marcus, Nov 12 2023

Status

approved