A335275 - OEIS

%I #56 Oct 09 2022 17:22:54

%S 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,

%T 29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,

%U 55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,73,74,75,76

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

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

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

%C Numbers whose powerful part (A057521) is a square.

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

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

%C Complement of A295661. - _Vaclav Kotesovec_, Jul 07 2020

%C 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

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

%H Amiram Eldar, <a href="/A335275/b335275.txt">Table of n, a(n) for n = 1..10000</a>

%H Eckford Cohen, <a href="https://doi.org/10.1090/S0002-9947-1964-0166181-5">Some asymptotic formulas in the theory of numbers</a>, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 3.1.2, p. 222.

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

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

%o (PARI) isok(k) = my(d=k/core(k)); gcd(d, k/d) == 1; \\ _Michel Marcus_, Jul 07 2020

%Y A000290, A138302 and A220218 are subsequences.

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

%K nonn

%O 1,2

%A _Amiram Eldar_, Jul 06 2020