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A337534
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Nontrivial squares together with nonsquares whose square part's square root is in the sequence.
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3
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4, 9, 16, 25, 32, 36, 48, 49, 64, 80, 81, 96, 100, 112, 121, 144, 160, 162, 169, 176, 196, 208, 224, 225, 240, 243, 256, 272, 289, 304, 324, 336, 352, 361, 368, 400, 405, 416, 441, 464, 480, 484, 486, 496, 512, 528, 529, 544, 560, 567, 576, 592, 608, 624, 625
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OFFSET
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1,1
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COMMENTS
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The appearance of a number is determined by its prime signature.
No terms are squarefree, as the square root of the square part of a squarefree number is 1.
If the square part of k is a 4th power, other than 1, k appears.
Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k is in this sequence if and only if there is m >= 1 such that column m of A329050 contains a member of S_k, but column m - 1 does not.
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LINKS
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FORMULA
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EXAMPLE
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4 is square and nontrivial (not 1), so 4 is in the sequence.
12 = 3 * 2^2 is nonsquare, but has square part 4, whose square root (2) is not in the sequence. So 12 is not in the sequence.
32 = 2 * 4^2 is nonsquare, and has square part 16, whose square root (4) is in the sequence. So 32 is in the sequence.
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MAPLE
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option remember ;
if n =1 then
4;
else
for a from procname(n-1)+1 do
return a;
end if;
end do:
end if;
end proc:
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MATHEMATICA
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pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[625], ! pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)
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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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