|
%I #41 Feb 16 2020 11:59:13
%S 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,
%T 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N Values of k in k-imperfect numbers.
%C The value of k in k-imperfect numbers, A127724. Except for the first term, all terms are > 1.
%C 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
%C 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
%H Michel Marcus, <a href="/A309806/a309806.gp.txt">PARI script</a>
%F a(n) = m/A206369(m) with m = A127724(n). - _M. F. Hasler_, Feb 14 2020
%e 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.
%t {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 *)
%o (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
%o lista(10^24) \\ to get 98 terms; _Michel Marcus_, Aug 20 2019
%o (PARI) A309806(n)=(n=A127724(n))/A206369(n) \\ _M. F. Hasler_, Feb 14 2020
%Y Cf. A127724, A127725 (2-imperfect), A127726 (3-imperfect), A206369 (rho).
%K nonn,more
%O 1,2
%A _Jud McCranie_, Aug 17 2019
%E a(45)-a(50) from _Giovanni Resta_, Aug 19 2019
|
