A309806 Values of k in k-imperfect numbers.

1, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Author

Jud McCranie, Aug 17 2019

Keywords

• nonn
• more

Comments

The value of k in k-imperfect numbers, A127724. Except for the first term, all terms are > 1.

It appears that the first instance of a(n)=4 is for n=98 (993803899780063855042560), while no integer is currently known to be 5-imperfect. - Michel Marcus, Aug 20 2019

A number n is called k-imperfect iff k := n/rho(n) is an integer, where rho = A206369 is a sum-of divisors function with alternating signs. - M. F. Hasler, Feb 14 2020

Examples

			The first 3 terms of A127724 are 1, 2, and 6, that are respectively 1-, 2-, and 3-imperfect. So the first 3 terms of this sequence are 1, 2 and 3.
		

Links

• Michel Marcus, PARI script

Crossrefs

Cf. A127724, A127725 (2-imperfect), A127726 (3-imperfect), A206369 (rho).

Programs

Mathematica

{1}~Join~Map[If[IntegerQ@ #, #, Nothing] &[#/Times @@ (Sum[(-1)^(#2 - k) #1^k, {k, 0, #2}] & @@@ FactorInteger[#])] &, Range[2, 10^6]] (* Michael De Vlieger, Feb 15 2020 *)

PARI

lista(lim) = {my(v = []); for (i=1, 4, my(vi = solveIMP(1, i, lim)); v = concat (v, vi);); apply(x->x/rhon(x), vecsort(v));} \\ uses the script in links section
lista(10^24) \\ to get 98 terms; Michel Marcus, Aug 20 2019

PARI

A309806(n)=(n=A127724(n))/A206369(n) \\ M. F. Hasler, Feb 14 2020

Formula

a(n) = m/A206369(m) with m = A127724(n). - M. F. Hasler, Feb 14 2020

Extensions

a(45)-a(50) from Giovanni Resta, Aug 19 2019