A127726 Numbers that are 3-imperfect.

6, 120, 126, 2520, 2640, 30240, 32640, 37800, 37926, 55440, 685440, 758520, 831600, 2600640, 5533920, 6917400, 9102240, 10281600, 11377800, 16687440, 152182800, 206317440, 250311600, 475917120, 866829600, 1665709920, 1881532800, 2082137400, 2147450880

Offset: 1

Author

T. D. Noe, Jan 25 2007

Keywords

• nonn

Comments

The new terms come from the paper by Zhou and Zhu. This sequence also contains n = 9223372034707292160 = 2^31*3*5*17*257*65537, which has the product of five Fermat primes (A019434). For this n, n/3 is a 2-imperfect number (A127725). - T. D. Noe, Apr 03 2009

From M. F. Hasler, Feb 13 2020: (Start)

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. (End)

Examples

			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.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..75 (terms < 10^20) (terms 1 to 35 from Donovan Johnson)
• Michel Marcus, More 3-imperfect numbers, (includes 15 terms beyond a(75)), Nov 23 2017.
• 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.

Crossrefs

Cf. A127724 (k-imperfect), A127725 (2-imperfect), A206369 (the rho function).

Programs

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 *)

PARI

isok(n) = 3*sumdiv(n, d, d*(-1)^bigomega(n/d)) == n; \\ Michel Marcus, Oct 28 2017

PARI

select( {is_A127726(n)=A206369(n)*3==n}, [1..10^5]*6) \\ M. F. Hasler, Feb 13 2020

Extensions

Extended by T. D. Noe, Apr 03 2009