OFFSET
1,1
COMMENTS
a(72) = 9223372034707292160 = 2^31*3*5*17*257*65537 has the product of five Fermat primes (A019434). Furthermore, a(72)/3 = A127725(10) is a 2-imperfect number. - T. D. Noe, Apr 03 2009; updated by Max Alekseyev, Oct 21 2025
By definition, n is k-imperfect iff n = k*A206369(n). So a k-imperfect number is always a multiple of k, and up to the first odd 3-imperfect number (larger than 10^49, if it exists, see Zhou & Zhu (2009)), all terms are a multiple of 6. - M. F. Hasler, Feb 13 2020
LINKS
Max Alekseyev and Michel Marcus, Table of n, a(n) for n = 1..90 (terms 1 to 35 from Donovan Johnson)
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.
Weiyi Zhou and Long Zhu, On k-imperfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.
EXAMPLE
6 = 2*3, so A206369(6) = (2 - 1)(3 - 1) = 2 = 6 / 3, so 6 is a term.
120 = 2^3 * 3 * 5, (8-4+2-1)*(3-1)*(5-1) = 40 = 120 / 3, so 120 is another term.
MATHEMATICA
okQ[n_] := 3 Sum[d*(-1)^PrimeOmega[n/d], {d, Divisors[n]}] == n;
For[k = 3, k < 10^6, k = k + 3, If[okQ[k], Print[k]]] (* Jean-François Alcover, Feb 01 2019 *)
PROG
(PARI) isok(n) = 3*sumdiv(n, d, d*(-1)^bigomega(n/d)) == n; \\ Michel Marcus, Oct 28 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Jan 25 2007
EXTENSIONS
a(26)-a(27) from Zhou and Zhu (2009) added by T. D. Noe, Apr 03 2009
Edited by Max Alekseyev, Oct 21 2025
STATUS
approved