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%I #24 Jun 24 2021 21:09:31
%S 0,0,0,1,2,1,4,4,4,5,8,5,10,9,8,11,14,10,16,13,14,17,20,15,20,21,20,
%T 21,26,19,28,26,26,29,28,25,34,33,32,31,38,31,40,37,34,41,44,37,44,42,
%U 44,45,50,43,48,47,50,53,56,45,58,57,52,57,58,55,64,61
%N a(n) = n + 1 - d(2n), where d(n) is the number of divisors of n.
%C Number of partitions of 2n into exactly two parts: (2n-i,i) such that i does not divide 2n-i. Complement of A066660.
%C Number of positive integers k <= n, such that k does not divide 2n-k. For example, a(12) = 5 since there are 5 positive integers k less than or equal to 12 that do not divide 2*12-k. They are 5, 7, 9, 10, and 11. - _Wesley Ivan Hurt_, Jun 24 2021
%H Muniru A Asiru, <a href="/A234306/b234306.txt">Table of n, a(n) for n = 1..5000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = n + 1 - A000005(2n).
%F a(n) = n - A066660(n).
%F a(n) = Sum_{i=1..n | i does not divide 2n-i} 1.
%e a(6) = 1; In this case, 2(6) = 12 has exactly 6 partitions into two parts: (11,1), (10,2), (9,3), (8,4), (7,5), (6,6). Note that 5 does not divide 7 but the smallest parts of the other partitions divide their corresponding largest parts. Therefore, a(6) = 1.
%p with(numtheory); A234306:=n->n + 1 - tau(2*n); seq(A234306(n), n=1..100);
%t Table[n + 1 - DivisorSigma[0, 2n], {n, 100}]
%o (PARI) a(n) = n + 1 - numdiv(2*n); \\ _Michel Marcus_, Dec 23 2013
%o (GAP) List([1..10^4], n->n+1-Tau(2*n)); # _Muniru A Asiru_, Feb 04 2018
%Y Cf. A000005, A066660.
%K nonn,easy
%O 1,5
%A _Wesley Ivan Hurt_, Dec 22 2013
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