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%I #8 Dec 09 2020 03:36:49
%S 0,1,1,1,1,2,2,4,4,7,7,11,11,19,18,27,28,43,42,61,63,92,92,127,130,
%T 187,188,252,263,355,364,483,503,665,683,883,925,1203,1248,1575,1663,
%U 2106,2202,2756,2904,3628,3813,4692,4965,6118,6458,7851,8342,10130,10717
%N Number of partitions of n into relatively prime parts where every part appears at least 2 times.
%C Moebius transform of A007690.
%H Vaclav Kotesovec, <a href="/A339244/b339244.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 1 + Sum_{n>=1} a(n) * x^n / (1 - x^n) = Product_{n>=1} (1 + x^(3*n)) / (1 - x^(2*n)).
%F a(n) = Sum_{d|n} mu(n/d) * A007690(d).
%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (6*sqrt(2)*n). - _Vaclav Kotesovec_, Dec 09 2020
%e a(10) = 7 because we have [4, 4, 1, 1], [3, 3, 2, 2], [3, 3, 1, 1, 1, 1], [2, 2, 2, 2, 1, 1], [2, 2, 2, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
%t A007690[n_] := SeriesCoefficient[Product[(1 + x^(3 k))/(1 - x^(2 k)), {k, 1, n}], {x, 0, n}]; a[n_] := Sum[MoebiusMu[n/d] A007690[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]
%Y Cf. A000837, A007690.
%K nonn
%O 1,6
%A _Ilya Gutkovskiy_, Nov 28 2020