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URL: https://oeis.org/A344256

⇱ A344256 - OEIS


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A344256
Number of partitions of n into 9 semiprime parts.
2
1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 5, 5, 8, 8, 10, 8, 12, 13, 18, 16, 19, 19, 24, 27, 31, 31, 37, 38, 46, 50, 57, 58, 65, 71, 80, 86, 96, 102, 115, 119, 134, 146, 160, 167, 181, 197, 217, 232, 252, 269, 290, 306, 333, 364, 387, 407, 434, 474, 512, 541
OFFSET
36,7
FORMULA
a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} [Omega(q) = Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p-q) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.
a(n) = [x^n y^9] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021
MATHEMATICA
h[n_] := h[n] = If[n == 0, 0, If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, If[i > n, 0, x*b[n-i, h[Min[n-i, i]]]]+b[n, h[i-1]]]], {x, 0, 10}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 9}];
Table[a[n], {n, 36, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz in A344257 *)
CROSSREFS
Cf. A001222 (Omega), A001358.
Column k=9 of A344447.
Sequence in context: A344246 A344254 A344255 * A344257 A088931 A088980
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, May 13 2021
EXTENSIONS
More terms from Alois P. Heinz, May 18 2021
STATUS
approved