1,1

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 A128608(n)=sigma(A128607(n)). Note that A128607(11)=1857437604=(2^2)*(3^2)*(11^2)*(653^2) has A128608(11)=5168743489=(7^3)*(13^3)*(19^3). Any other cubes or higher egcd's in A128608? Conjecture II: egcd(A128608(n))=2 for all n ne 11.

A128608(2)=sigma(A128607(2))=sigma(343)=1+7+7^2+7^3=400=2^4*5^2.

with(numtheory); egcd:=proc(n::posint) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z-> z[2], L); return igcd(op(L)) fi end: # egcd(1) is NULL M:=[]: for w to 1 do for n from 2 do if egcd(n)>1 then sn:=sigma(n); if egcd(sn)>1 then M:=[op(M), n]; print(ifactor(n), ifactor(sn)) fi fi; od od; #This is a simplified program. Loop over powers of n if you want. Another way is to loop over powers of p, p*q, p*q*r out to p*q*r*s*t.

Cf. A000203, A001597, A128607.

Sequence in context: A112075 A068872 A167721 this_sequence A144719 A017282 A036304

Adjacent sequences: A128605 A128606 A128607 this_sequence A128609 A128610 A128611

nonn

Walter A. Kehowski (wkehowski(AT)cox.net), Mar 20 2007