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

1,2

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

Links

Table of n, a(n) for n=1..50.

Michel Marcus, PARI script

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A281976 A300708 A240755 * A256170 A051686 A079294

Adjacent sequences: A309803 A309804 A309805 * A309807 A309808 A309809

Keyword

nonn,more

Author

Jud McCranie, Aug 17 2019

Extensions

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

Status

approved