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 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. 3
 1, 121, 400, 961, 116281, 2989441, 7958041, 361722361, 962922961, 1902442689, 1891467081, 5168743489, 4755619521, 5215583961, 6835486329, 7496615889, 13884144561, 13884144561, 35018011161, 120776405841, 120776405841, 230195565369, 253358202409, 171651947481 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. LINKS Robert Israel, Table of n, a(n) for n = 1..58 EXAMPLE a(2) = sigma(A128607(2)) = sigma(343) = 1+7+7^2+7^3 = 400 = 2^4*5^2. 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 *) CROSSREFS Cf. A000203, A001597, A128607. Sequence in context: A253321 A253328 A257035 * A326710 A144719 A222551 Adjacent sequences: A128605 A128606 A128607 * A128609 A128610 A128611 KEYWORD nonn AUTHOR Walter Kehowski, Mar 20 2007 EXTENSIONS 1, 13884144561, 35018011161, 120776405841, added by Zak Seidov, Feb 14 2016 Edited by Robert Israel, Feb 14 2016 STATUS approved

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Last modified April 23 13:11 EDT 2024. Contains 371913 sequences. (Running on oeis4.)