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A200928 Number of partitions of n such that the number of parts and the smallest part are coprime. 3

%I

%S 1,2,3,4,6,10,14,19,26,37,49,67,89,120,158,206,267,346,443,565,718,

%T 908,1142,1431,1785,2221,2749,3394,4175,5128,6271,7652,9311,11306,

%U 13687,16538,19931,23977,28776,34470,41203,49173,58559,69625,82630,97913,115816

%N Number of partitions of n such that the number of parts and the smallest part are coprime.

%H Alois P. Heinz, <a href="/A200928/b200928.txt">Table of n, a(n) for n = 1..400</a>

%F a(n) = A000041(n) - A201025(n).

%e a(4) = 4: [1,1,1,1], [1,1,2], [1,3], [4];

%e a(5) = 6: [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [1,4], [5];

%e a(6) = 10: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2,3],[3,3], [1,1,4], [1,5], [6].

%p with (combinat):

%p b:= proc(n, j, t, s) option remember;

%p add (b(n-i, i, irem(t+1,s), s), i=j..iquo(n, 2))+

%p `if`(igcd(t, s)=1, 1, 0)

%p end:

%p a:= n-> numbpart(n-1) +`if`(n>1, 1, 0) +add (b(n-i, i, 2, i), i=2..iquo(n, 2)):

%p seq (a(n), n=1..50);

%t b[n_, j_, t_, s_] := b[n, j, t, s] = Sum[b[n-i, i, Mod[t+1, s], s], {i, j, Quotient[n, 2]}] + If[GCD[t, s]==1, 1, 0]; a[n_] := PartitionsP[n-1] + If[n>1, 1, 0] + Sum [b[n-i, i, 2, i], {i, 2, Quotient[n, 2]}]; Table[a[n], {n, 1, 50}] (* _Jean-Fran├žois Alcover_, Feb 15 2017, translated from Maple *)

%Y Cf. A000041, A199888, A201025.

%K nonn

%O 1,2

%A _Alois P. Heinz_, Nov 25 2011

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Last modified June 15 07:12 EDT 2021. Contains 345043 sequences. (Running on oeis4.)