

A335740


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 not a power of 2.


5



3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 90
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OFFSET

1,1


COMMENTS

Every missing number greater than 2 is a multiple of 4. Every power of 2 is missing. Every positive power of every squarefree number greater than 2 is present.
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 greater than 2. k can be shown to be A299090(m).


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 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 not 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 in the sequence.


MATHEMATICA

f[p_, e_] := p^Floor[e/2]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2, 100], FixedPointList[s, #] [[3]] > 2 &] (* Amiram Eldar, Nov 27 2020 *)


CROSSREFS

Complement within A020725 of A335738.
A000188, A299090 are used in a formula defining this sequence.
Powers of squarefree numbers: A005117(1), A144338(1), A062503(2), A113849(4).
Subsequences: A042968\{1,2}, A182853, A268390.
With {1}, numbers in the odd bisection of A336322.
Sequence in context: A116883 A256543 A186145 * A352873 A047984 A288513
Adjacent sequences: A335737 A335738 A335739 * A335741 A335742 A335743


KEYWORD

nonn


AUTHOR

Peter Munn, Jun 20 2020


STATUS

approved



