A090477 T(n,k) is the number of occurrences of the k-th prime in the first n partition numbers (with repetitions), 1 <= k <= n, triangular array read by rows.

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 2, 0, 0, 0, 3, 3, 3, 1, 2, 0, 0, 0, 0, 4, 4, 3, 2, 2, 0, 0, 0, 0, 0, 7, 4, 3, 3, 2, 0, 0, 0, 0, 0, 0, 7, 4, 3, 4, 3, 0, 0, 0, 0, 0, 0, 0, 7, 4, 3, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 7, 7, 4, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,23

Links

Table of n, a(n) for n=1..105.

Example

Prime factorizations of the first 12 partition numbers:
p(1)=1, p(2)=2, p(3)=3, p(4)=5, p(5)=7, p(6)=11, p(7)=15=5*3,
p(8)=22=11*2, p(9)=30=5*3*2, p(10)=42=7*3*2 and p(11)=56=7*2^3:
therefore T(11,1)=0+1+0+0+0+0+0+1+1+1+3=7 and T(11,2)=0+0+1+0+0+0+1+0+1+1+0=4.
Triangle begins:
  0;
  1, 0;
  1, 1, 0;
  1, 1, 1, 0;
  1, 1, 1, 1, 0;
  1, 1, 1, 1, 1, 0;
  1, 2, 2, 1, 1, 0, 0;
  ...

Maple

b:= proc(n) option remember; `if`(n=0, 0, b(n-1)+add(i[2]*
     x^numtheory[pi](i[1]), i=ifactors(combinat[numbpart](n))[2]))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Sep 19 2021

Mathematica

T[n_, k_] := Select[FactorInteger /@ PartitionsP[Range[0, n]] // Flatten[#, 1]&, #[[1]] == Prime[k]&][[All, 2]] // Total;
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 18 2021 *)

Prog

(PARI) T(n, k) = sum(i=1, n, valuation(numbpart(i), prime(k))); \\ Michel Marcus, Sep 18 2021

Crossrefs

Cf. A000040, A000041.

Sequence in context: A359216 A274370 A128521 * A349802 A368753 A345907

Adjacent sequences: A090474 A090475 A090476 * A090478 A090479 A090480

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Feb 01 2004

Extensions

Name edited by Jean-François Alcover, Sep 18 2021

Status

approved