A128608 - OEIS

%I #18 Feb 09 2023 07:25:21

%S 1,121,400,961,116281,2989441,7958041,361722361,962922961,1902442689,

%T 1891467081,5168743489,4755619521,5215583961,6835486329,7496615889,

%U 13884144561,13884144561,35018011161,120776405841,120776405841,230195565369,253358202409,171651947481

%N a(n)=sigma(A128607(n)), where A128607(n) is the sequence of perfect (or pure) powers such that a(n) is a perfect power.

%C Denote by egcd(n) the gcd of all the powers in the prime factorization of n. In our context, a square has egcd=2, a cube has egcd=3 and so on. The only elements n in the sequence for which egcd(n)>2 are 81 and 343. Are there any others? Conjecture I: egcd(A128607(n))=2 for all n>2. Let a(n)=sigma(A128607(n)). Note that A128607(11)=1857437604=(2^2)*(3^2)*(11^2)*(653^2) has a(11)=5168743489=(7^3)*(13^3)*(19^3). Any other cubes or higher egcd's in this sequence? Conjecture II: egcd(a(n))=2 for all n ne 11.

%H Robert Israel, <a href="/A128608/b128608.txt">Table of n, a(n) for n = 1..58</a>

%e a(2) = sigma(A128607(2)) = sigma(343) = 1+7+7^2+7^3 = 400 = 2^4*5^2.

%p N:= 10^13: # to get all terms where A128607(n) <= N

%p pows:= {1, seq(seq(n^k, n = 2 .. floor(N^(1/k))), k = 2 .. floor(log[2](N)))}:

%p filter:= proc(n) local s, F;

%p    s:= numtheory:-sigma(n);

%p    F:= map(t -> t[2], ifactors(s)[2]);

%p    igcd(op(F)) >= 2

%p end proc:

%p filter(1):= true: A128608:= sort(convert((filter, pows), list)):

%p map(numtheory:-sigma,A128608); # _Robert Israel_, Feb 14 2016

%t M = 10^13; (* to get all terms where A128607(n) <= M *)

%t pows = {1, Table[Table[n^k, {n, 2, Floor[M^(1/k)]}], {k, 2, BitLength[M]-1}]} // Flatten // Union;

%t okQ[n_] := Module[{s, F}, s = DivisorSigma[1, n]; F = FactorInteger[s][[All, 2]]; GCD @@ F >= 2];

%t okQ[1] = True;

%t DivisorSigma[1, #]& /@ Select[pows, okQ] (* _Jean-François Alcover_, Feb 09 2023, after _Robert Israel_ *)

%Y Cf. A000203, A001597, A128607.

%K nonn

%O 1,2

%A _Walter Kehowski_, Mar 20 2007

%E 1, 13884144561, 35018011161, 120776405841, added by _Zak Seidov_, Feb 14 2016

%E Edited by _Robert Israel_, Feb 14 2016