A127725 Numbers that are 2-imperfect.

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

Offset: 1

Author

T. D. Noe, Jan 25 2007

Keywords

• nonn
• more
• hard

Comments

This sequence also contains n = 3074457344902430720 = 2^31*5*17*257*65537, which has the product of four Fermat primes (A019434). For this n, 3*n is a 3-imperfect number (A127726). - T. D. Noe, Apr 03 2009

a(9) > 2*10^11. - Donovan Johnson, Feb 07 2013

62549517598720 is also a term (see the "43 terms > 2*10^11" link by Donovan Johnson in A127724). - Michel Marcus, Nov 05 2017

Examples

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

Links

• 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.

Crossrefs

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

Programs

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 *)

PARI

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

Extensions

a(9) by Jud McCranie, Aug 17 2019