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%I #10 Nov 22 2016 10:03:42
%S 0,0,0,0,4,0,6,4,8,9,10,4,6,6,4,12,0,8,10,8,6,10,18,16,21,13,16,27,24,
%T 19,27,8,10,26,29,18,25,16,0,28,27,6,32,21,19,33,13,13,13,21,0,18,23,
%U 48,28,16,26,34,26,4,33,35,4,40,52,10,65,34,62,16,40,12,66,48,21,18,6,13,72,46,19,20
%N Last residue class to appear in the partition numbers mod n.
%C Let k be the smallest number such that {p(1), p(2), ..., p(k)} contains each residue class mod n, where p(m) is the m-th partition number A000041(m). Then a(n) = p(k) mod n.
%H Charles R Greathouse IV, <a href="/A278210/b278210.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000041(A278209(n)) mod n.
%e p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(5) = 7, p(6) = 11, p(7) = 15, p(8) = 22, p(9) = 30, p(10) = 42, and p(11) = 56. Mod 4 these are 1, 2, 3, 1, 3, 3, 3, 2, 2, 2, and 0 respectively. The last term to appear is 0 mod 4 at index 11, so a(4) = 0.
%t Table[k = 1; While[Length@ Union@ Map[Mod[PartitionsP@ #, n] &, Range@ k] != n, k++]; Mod[PartitionsP@ k, n], {n, 82}] (* _Michael De Vlieger_, Nov 21 2016 *)
%o (PARI) a(n)=if(n==1, return(1)); my(v=vectorsmall(n),s=1,t,k=1); v[2]=1; while(s<n,t=numbpart(k++)%n; if(v[t+1]==0, v[t+1]=1; if(s++==n, return(t))))
%Y Cf. A000041, A278209.
%K nonn
%O 1,5
%A _Charles R Greathouse IV_, Nov 15 2016
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