A278210 Last residue class to appear in the partition numbers mod n.

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, 19, 27, 8, 10, 26, 29, 18, 25, 16, 0, 28, 27, 6, 32, 21, 19, 33, 13, 13, 13, 21, 0, 18, 23, 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

Offset

1,5

Comments

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.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000041(A278209(n)) mod n.

Example

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.

Mathematica

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 *)

Prog

(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))))

Crossrefs

Cf. A000041, A278209.

Sequence in context: A327278 A374970 A374969 * A291540 A075443 A021250

Adjacent sequences: A278207 A278208 A278209 * A278211 A278212 A278213

Keyword

nonn

Author

Charles R Greathouse IV, Nov 15 2016

Status

approved