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A374291
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Squares of powerful numbers.
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2
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1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 5184, 6561, 10000, 11664, 14641, 15625, 16384, 20736, 28561, 38416, 40000, 46656, 50625, 59049, 65536, 82944, 83521, 104976, 117649, 130321, 153664, 160000, 186624, 194481, 234256, 250000, 262144, 279841, 331776
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OFFSET
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1,2
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COMMENTS
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First differs from A340588 at n = 12.
4-full (or 3-full) squares.
Numbers whose exponents in their prime factorization are all even numbers >= 4.
This sequence is closed under multiplication.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p^2-1))) = zeta(4)*zeta(6)/zeta(12) = 15015/(1382*Pi^2) = 1.10082313486953808844... .
Sum_{n>=1} 1/a(n)^s = Product_{p prime} (1 + 1/(p^(2*s)*(p^(2*s)-1))) = zeta(4*s)*zeta(6*s)/zeta(12*s), for s > 1/4.
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MATHEMATICA
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powQ[n_] := n==1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[600], powQ]^2
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PROG
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(PARI) is(k) = issquare(k) && ispowerful(sqrtint(k));
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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