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A090477
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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.
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0
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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
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OFFSET
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1,23
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LINKS
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EXAMPLE
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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;
...
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MAPLE
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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)):
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MATHEMATICA
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T[n_, k_] := Select[FactorInteger /@ PartitionsP[Range[0, n]] // Flatten[#, 1]&, #[[1]] == Prime[k]&][[All, 2]] // Total;
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PROG
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(PARI) T(n, k) = sum(i=1, n, valuation(numbpart(i), prime(k))); \\ Michel Marcus, Sep 18 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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