OFFSET
1,23
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
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Feb 01 2004
EXTENSIONS
Name edited by Jean-François Alcover, Sep 18 2021
STATUS
approved