A393940 Numbers m such that tau(m) is a perfect number (A000396).

12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428

Offset

1,1

Comments

Sequence deviates from A030515 and A162947, the smallest term that is not term of these sequences is 960.

Conjecture: number m = 18 is only number such that tau(m) and phi(m) are both perfect numbers.

This is the sequence of the smallest numbers m such that tau(m) = A000396(n): 12, 960, 1014686023680 = A081620(31), A081620(127), ...

Conjecture is false: with m = 2^61 * 17 * 65537 * ((2^2281-1)*2^2200+1), tau(m) and phi(m) are both perfect numbers. - Michel Marcus, Mar 12 2026

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

factordb, (2^2281-1)*2^2200+1.

Example

Number 12 is in the sequence because tau(12) = 6 (perfect number).

Mathematica

Select[Range[500], DivisorSigma[-1, DivisorSigma[0, #]] == 2 &] (* Amiram Eldar, Mar 03 2026 *)

Prog

(Magma) [m: m in [1..500] | &+Divisors(#Divisors(m)) eq 2 * #Divisors(m)];
(PARI) isok(m) = my(d=numdiv(m)); sigma(d) == 2*d; \\ Michel Marcus, Mar 12 2026

Crossrefs

Cf. A146542 (sigma(m) is a perfect number), A393942 (phi(m) is a perfect number).

Cf. A000005 (tau), A000010 (phi), A000396, A020488, A081357 (sublime numbers).

Cf. A030515 (subsequence), A137491 (subsequence).

Sequence in context: A253388 A030515 A162947 * A351201 A391023 A359892

Adjacent sequences: A393937 A393938 A393939 * A393941 A393942 A393943

Keyword

nonn

Author

Jaroslav Krizek, Mar 03 2026

Status

approved