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%I #73 May 17 2024 06:51:55
%S 1,5,7,11,13,17,19,23,29,31,35,37,41,43,47,53,55,59,61,65,67,71,73,77,
%T 79,83,85,89,91,95,97,101,103,107,109,113,115,119,127,131,133,137,139,
%U 143,145,149,151,155,157,161,163,167,173,179,181,185,187,191,193,197,199,203,205,209,211
%N Numbers k such that 6*k is squarefree.
%C These are the numbers from A005117 that are not divisible by 2 and 3.
%C Squarefree numbers coprime to 6. - _Robert Israel_, Sep 02 2016
%C Numbers k such that A008588(k) is in A005117. - _Felix Fröhlich_, Sep 02 2016
%C The asymptotic density of this sequence is 3/Pi^2 (A104141). - _Amiram Eldar_, May 22 2020
%C From _Peter Munn_, Nov 20 2020: (Start)
%C The products generated from each subset of A215848 (primes greater than 3).
%C Closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897. (End)
%C Multiplied by 6 we have 6, 30, 42, 66, 78, 102, ..., the values that may appear in A076978 after the 1, 2. [_Don Reble_, Dec 02 2020] - _R. J. Mathar_, Dec 15 2020
%C By the von Staudt-Clausen theorem, denominators of Bernoulli numbers are of the form 6*a(n) for some n. - _Charles R Greathouse IV_, May 16 2024
%H Robert Israel, <a href="/A276378/b276378.txt">Table of n, a(n) for n = 1..10000</a>
%F {a(n) : n >= 1} = {A003961(A003961(A005117(n))) : n >= 1} = {A003961(A056911(n))) : n >= 1}. - _Peter Munn_, Nov 20 2020
%F Sum_{n>=1} 1/a(n)^s = (6^s)*zeta(s)/((1+2^s)*(1+3^s)*zeta(2*s)), s>1. - _Amiram Eldar_, Sep 26 2023
%e 5 is in this sequence because 6*5 = 30 = 2*3*5 is squarefree.
%p select(numtheory:-issqrfree, [seq(seq(6*i+j,j=[1,5]),i=0..100)]); # _Robert Israel_, Sep 02 2016
%t Select[Range@ 212, SquareFreeQ[6 #] &] (* _Michael De Vlieger_, Sep 02 2016 *)
%o (Magma) [n: n in [1..230] | IsSquarefree(6*n)];
%o (PARI) is(n) = issquarefree(6*n) \\ _Felix Fröhlich_, Sep 02 2016
%Y Numbers m such that k*m is squarefree: A005117 (k = 1), A056911 (k = 2), A261034 (k = 3), A274546 (k = 5).
%Y Subsequence of A007310, A300957, and A339690.
%Y Cf. A003961, A008588, A059897, A076978, A104141, A215848.
%K nonn,easy
%O 1,2
%A _Juri-Stepan Gerasimov_, Sep 02 2016
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