A337534 Nontrivial squares together with nonsquares whose square part's square root is in the sequence.

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

Offset: 1

Author

Peter Munn, Aug 31 2020

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Square part
• Index to sequences related to prime signature

Crossrefs

Complement of A337533.

Subsequences: A000290\{0,1}, A082294.

Subsequence of: A013929, A162643.

A209229, A267116 are used in a formula defining this sequence.

Cf. A008833, A050376, A329050.

Programs

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 *)

Formula

Numbers k such that A209229(A267116(k) + 1) = 0.

A008833(a(n)) > 1.