

A335738


Factorize each integer m >= 2 as the product of powers of nonunit squarefree numbers with distinct exponents that are powers of 2. The sequence lists m such that the factor with the largest exponent is a power of 2.


5



2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 60, 64, 68, 76, 80, 84, 88, 92, 96, 104, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 184, 188, 192, 204, 208, 212, 220, 224, 228, 232, 236, 240, 244, 248, 256, 260, 264, 268, 272
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OFFSET

1,1


COMMENTS

2 is the only term not divisible by 4. All powers of 2 are present. Every term divisible by an odd square is divisible by 16, the first such being 144.
The defined factorization is unique. Every positive number is a product of at most one squarefree number (A005117), at most one square of a squarefree number (A062503), at most one 4th power of a squarefree number (A113849), at most one 8th power of a squarefree number, and so on.
Iteratively map m using A000188, until 1 is reached, as A000188^k(m), for some k >= 1. m is in the sequence if and only if the preceding number, A000188^(k1)(m), is 2. k can be shown to be A299090(m).
Closed under squaring, but not closed under multiplication: 12 = 3^1 * 2^2 and 432 = 3^1 * 3^2 * 2^4 are in the sequence, but 12 * 432 = 5184 = 3^4 * 2^6 = 2^2 * 6^4 is not.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000


FORMULA

{a(n)} = {m : m >= 2 and A000188^(k1)(m) = 2, where k = A299090(m)}.


EXAMPLE

6 is a squarefree number, so its factorization for the definition (into powers of nonunit squarefree numbers with distinct exponents that are powers of 2) is the trivial "6^1". 6^1 is therefore the factor with the largest exponent, and is not a power of 2, so 6 is not in the sequence.
48 factorizes for the definition as 3^1 * 2^4. The factor with the largest exponent is 2^4, which is a power of 2, so 48 is in the sequence.
10^100 (a googol) factorizes in this way as 10^4 * 10^32 * 10^64. The factor with the largest exponent, 10^64, is a power of 10, not a power of 2, so 10^100 is not in the sequence.


MATHEMATICA

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 300], FixedPointList[s, #] [[3]] == 2 &] (* Amiram Eldar, Nov 27 2020 *)


CROSSREFS

Complement within A020725 of A335740.
A000188, A299090 are used in a formula defining this sequence.
Powers of squarefree numbers: A005117(1), A144338(1), A062503(2), A113849(4).
Subsequences: A000079\{1}, A001749, A181818\{1}, A273798.
Sequence in context: A215459 A019442 A048166 * A010066 A180490 A160408
Adjacent sequences: A335735 A335736 A335737 * A335739 A335740 A335741


KEYWORD

nonn


AUTHOR

Peter Munn, Jun 20 2020


STATUS

approved



