A127725 2-imperfect numbers.

2, 12, 40, 252, 880, 10880, 75852, 715816960, 62549517598720, 3074457344902430720

Offset

1,1

Comments

a(10) = 2^31*5*17*257*65537 contains the product of four Fermat primes (A019434). If a term m is coprime to 3, then 3*m is a 3-imperfect number (A127726). - T. D. Noe, Apr 03 2009; updated by Max Alekseyev, Oct 21 2025

Links

Table of n, a(n) for n=1..10.

Laszlo Toth, The alternating sum-of-divisors function, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siofok, Hungary.

Laszlo Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Example

40 = 2^3 * 5, (8 - 4 + 2 - 1)(5 - 1) = 20 = 40 / 2, so 40 is in the sequence. - Jud McCranie, Aug 17 2019

Mathematica

okQ[n_] := 2 Sum[d*(-1)^PrimeOmega[n/d], {d, Divisors[n]}] == n;
For[k = 2, k <= 10^9, k = k+2, If[okQ[k], Print[k]]] (* Jean-François Alcover, Jan 27 2019 *)

Prog

(PARI) isok(n) = 2*sumdiv(n, d, d*(-1)^bigomega(n/d)) == n; \\ Michel Marcus, Oct 28 2017

Crossrefs

Cf. A127726 (3-imperfect numbers), A328860 (4-imperfect numbers), A127724 (k-imperfect numbers).

Sequence in context: A110953 A188572 A098519 * A371357 A280174 A185619

Adjacent sequences: A127722 A127723 A127724 * A127726 A127727 A127728

Keyword

nonn,more,hard

Author

T. D. Noe, Jan 25 2007

Extensions

a(9) from Michel Marcus confirmed and added by Jud McCranie, Aug 17 2019

a(10) from T. D. Noe confirmed and added by Max Alekseyev, Oct 21 2025

Status

approved