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A214681 a(n) is obtained from n by removing factors of 2 and 3 that do not contribute to a factor of 6.

%I A214681 #43 Feb 19 2025 02:44:54
%S A214681 1,1,1,1,5,6,7,1,1,5,11,6,13,7,5,1,17,6,19,5,7,11,23,6,25,13,1,7,29,
%T A214681 30,31,1,11,17,35,36,37,19,13,5,41,42,43,11,5,23,47,6,49,25,17,13,53,
%U A214681 6,55,7,19,29,59,30,61,31,7,1,65,66,67,17,23,35,71,36
%N A214681 a(n) is obtained from n by removing factors of 2 and 3 that do not contribute to a factor of 6.
%C A214681 In this sequence, the number 6 exhibits some characteristics of a prime number since we have removed extraneous 2's and 3's from the prime factorizations of numbers.
%H A214681 Alois P. Heinz, <a href="/A214681/b214681.txt">Table of n, a(n) for n = 1..10000</a>
%F A214681 a(n) = n*6^(v_6(n))/(2^(v_2(n))*3^(v_3(n))), where v_k(n) is the k-adic valuation of n, that is v_k(n) gives the largest power of k, a, such that k^a divides n.
%F A214681 Sum_{k=1..n} a(k) ~ (7/24) * n^2. - _Amiram Eldar_, Dec 25 2023
%e A214681 For n=4, v_2(4)=2, v_3(4)=0, and v_6(4)=0, so a(4) = 4*1/(4*1) = 1.
%e A214681 For n=36, v_2(36)=2, v_3(36)=2, and v_6(36)=2, so a(36) = 36*36/(4*9) = 36.
%e A214681 For n=17, a(17) = 17 since 17 has no factors of 6, 2 or 3.
%p A214681 a:= proc(n) local i, m, r; m:=n;
%p A214681       for i from 0 while irem(m, 6, 'r')=0 do m:=r od;
%p A214681       while irem(m, 2, 'r')=0 do m:=r od;
%p A214681       while irem(m, 3, 'r')=0 do m:=r od;
%p A214681       m*6^i
%p A214681     end:
%p A214681 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jul 04 2013
%t A214681 With[{v = IntegerExponent}, a[n_] := n*6^v[n, 6]/2^v[n, 2]/3^v[n, 3]; Array[a, 100]] (* _Amiram Eldar_, Dec 09 2020 *)
%o A214681 (Sage)
%o A214681 n=100 #change n for more terms
%o A214681 C=[]
%o A214681 b=6
%o A214681 P = factor(b)
%o A214681 for i in [1..n]:
%o A214681     prod = 1
%o A214681     for j in range(len(P)):
%o A214681         prod = prod * ((P[j][0])^(Integer(i).valuation(P[j][0])))
%o A214681     C.append((b^(Integer(i).valuation(b)) * i) /prod)
%Y A214681 Cf. A007814, A007949, A122841.
%Y A214681 Cf. A214682, A214685.
%K A214681 easy,nonn
%O A214681 1,5
%A A214681 _Tom Edgar_, Jul 25 2012