A128608 a(n)=sigma(A128607(n)), where A128607(n) is the sequence of perfect (or pure) powers such that a(n) is a perfect power.

1, 121, 400, 961, 116281, 2989441, 7958041, 361722361, 962922961, 1902442689, 1891467081, 5168743489, 4755619521, 5215583961, 6835486329, 7496615889, 13884144561, 13884144561, 35018011161, 120776405841, 120776405841, 230195565369, 253358202409, 171651947481

Offset: 1

Author

Walter Kehowski, Mar 20 2007

Keywords

• nonn

Comments

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.

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..58

Crossrefs

Cf. A000203, A001597, A128607.

Programs

Maple

N:= 10^13: # to get all terms where A128607(n) <= N
pows:= {1, seq(seq(n^k, n = 2 .. floor(N^(1/k))), k = 2 .. floor(log[2](N)))}:
filter:= proc(n) local s, F;
   s:= numtheory:-sigma(n);
   F:= map(t -> t[2], ifactors(s)[2]);
   igcd(op(F)) >= 2
end proc:
filter(1):= true: A128608:= sort(convert((filter, pows), list)):
map(numtheory:-sigma,A128608); # Robert Israel, Feb 14 2016

Mathematica

M = 10^13; (* to get all terms where A128607(n) <= M *)
pows = {1, Table[Table[n^k, {n, 2, Floor[M^(1/k)]}], {k, 2, BitLength[M]-1}]} // Flatten // Union;
okQ[n_] := Module[{s, F}, s = DivisorSigma[1, n]; F = FactorInteger[s][[All, 2]]; GCD @@ F >= 2];
okQ[1] = True;
DivisorSigma[1, #]& /@ Select[pows, okQ] (* Jean-François Alcover, Feb 09 2023, after Robert Israel *)

Extensions

1, 13884144561, 35018011161, 120776405841, added by Zak Seidov, Feb 14 2016

Edited by Robert Israel, Feb 14 2016