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A337534 Nontrivial squares together with nonsquares whose square part's square root is in the sequence. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Eric Weisstein's World of Mathematics, Square part
FORMULA
Numbers k such that A209229(A267116(k) + 1) = 0.
A008833(a(n)) > 1.
EXAMPLE
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.
MAPLE
A337534 := proc(n)
option remember ;
if n =1 then
4;
else
for a from procname(n-1)+1 do
if A209229(A267116(a)+1) = 0 then
return a;
end if;
end do:
end if;
end proc:
seq(A337534(n), n=1..80) ; # R. J. Mathar, Feb 16 2021
MATHEMATICA
pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[625], ! pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)
CROSSREFS
Complement of A337533.
Subsequences: A000290\{0,1}, A082294.
Subsequence of: A013929, A162643.
A209229, A267116 are used in a formula defining this sequence.
Sequence in context: A010433 A175592 A331219 * A126589 A340585 A010409
KEYWORD
nonn,easy
AUTHOR
Peter Munn, Aug 31 2020
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)