A335275 Numbers k such that the largest square dividing k is a unitary divisor of k.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Author

Amiram Eldar, Jul 06 2020

Keywords

• nonn

Comments

Numbers k such that gcd(A008833(k), k/A008833(k)) = 1.

Numbers whose prime factorization contains exponents that are either 1 or even.

Numbers whose powerful part (A057521) is a square.

First differs from A220218 at n = 227: a(227) = 256 is not a term of A220218.

The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465).

Complement of A295661. - Vaclav Kotesovec, Jul 07 2020

Differs from A096432 in having or not having 1, 256, 432, 648, 768, 1280, 1728, 1792, 2000, 2160, 2304,... - R. J. Mathar, Jul 22 2020

Equivalently, numbers k whose squarefree part (A007913) is a unitary divisor, or gcd(A007913(k), A008833(k)) = 1. - Amiram Eldar, Oct 09 2022

Examples

			12 is a term since the largest square dividing 12 is 4, and 4 and 12/4 = 3 are coprime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 3.1.2, p. 222.

Crossrefs

A000290, A138302 and A220218 are subsequences.

Cf. A007913, A008833, A057521, A065465, A295661.

Programs

Mathematica

seqQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 1 || EvenQ[#] &];  Select[Range[100], seqQ]

PARI

isok(k) = my(d=k/core(k)); gcd(d, k/d) == 1; \\ Michel Marcus, Jul 07 2020