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%I #14 Feb 16 2021 13:15:57
%S 4,9,16,25,32,36,48,49,64,80,81,96,100,112,121,144,160,162,169,176,
%T 196,208,224,225,240,243,256,272,289,304,324,336,352,361,368,400,405,
%U 416,441,464,480,484,486,496,512,528,529,544,560,567,576,592,608,624,625
%N Nontrivial squares together with nonsquares whose square part's square root is in the sequence.
%C The appearance of a number is determined by its prime signature.
%C No terms are squarefree, as the square root of the square part of a squarefree number is 1.
%C If the square part of k is a 4th power, other than 1, k appears.
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePart.html">Square part</a>
%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F Numbers k such that A209229(A267116(k) + 1) = 0.
%F A008833(a(n)) > 1.
%e 4 is square and nontrivial (not 1), so 4 is in the sequence.
%e 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.
%e 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.
%p A337534 := proc(n)
%p option remember ;
%p if n =1 then
%p 4;
%p else
%p for a from procname(n-1)+1 do
%p if A209229(A267116(a)+1) = 0 then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc:
%p seq(A337534(n),n=1..80) ; # _R. J. Mathar_, Feb 16 2021
%t pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[625], ! pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* _Amiram Eldar_, Sep 18 2020 *)
%Y Complement of A337533.
%Y Subsequences: A000290\{0,1}, A082294.
%Y Subsequence of: A013929, A162643.
%Y A209229, A267116 are used in a formula defining this sequence.
%Y Cf. A008833, A050376, A329050.
%K nonn,easy
%O 1,1
%A _Peter Munn_, Aug 31 2020
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