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A128608
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A128608(n)=sigma(A128607(n)), where A128607(n) is the sequence of perfect (or pure) powers such that A128608(n) is a perfect power.
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3
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121, 400, 961, 116281, 2989441, 7958041, 361722361, 962922961, 1902442689, 1891467081, 5168743489, 4755619521, 5215583961, 6835486329, 7496615889, 13884144561, 230195565369, 228867516801, 575429962041, 631085659281
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 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.
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EXAMPLE
| A128608(2)=sigma(A128607(2))=sigma(343)=1+7+7^2+7^3=400=2^4*5^2.
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MAPLE
| 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.
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CROSSREFS
| Cf. A000203, A001597, A128607.
Sequence in context: A068872 A203856 A167721 * A144719 A017282 A036304
Adjacent sequences: A128605 A128606 A128607 * A128609 A128610 A128611
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KEYWORD
| nonn
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AUTHOR
| Walter A. Kehowski (wkehowski(AT)cox.net), Mar 20 2007
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