A340756 -id:A340756 - cp's OEIS frontend

A344447 Number T(n,k) of partitions of n into k semiprimes; triangle T(n,k), n>=0, read by rows.

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

Offset: 0

Alois P. Heinz, May 19 2021

T(n,k) is defined for all n,k >= 0. The triangle contains in each row n only the terms for k=0 and then up to the last positive T(n,k) (if it exists).

			Triangle T(n,k) begins:
  1 ;
  0 ;
  0 ;
  0 ;
  0, 1 ;
  0    ;
  0, 1 ;
  0    ;
  0, 0, 1 ;
  0, 1    ;
  0, 1, 1 ;
  0       ;
  0, 0, 1, 1 ;
  0, 0, 1    ;
  0, 1, 1, 1 ;
  0, 1, 1    ;
  0, 0, 1, 1, 1 ;
  0, 0, 0, 1    ;
  0, 0, 2, 2, 1 ;
  0, 0, 2, 1    ;
  0, 0, 2, 1, 1, 1 ;
  ...

Alois P. Heinz, Rows n = 0..500, flattened

Columns k=0-10 give: A000007, A064911, A072931, A344446, A340756, A344245, A344246, A344254, A344255, A344256, A344257.
Row sums give A101048.
T(4n,n) gives A000012.
Cf. A001358, A117278, A281617.

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     `if`(i>n, 0, expand(x*b(n-i, h(min(n-i, i)))))+b(n, h(i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n, h(n))):
seq(T(n), n=0..32);

h[n_] := h[n] = If[n == 0, 0,
     If[PrimeOmega[n] == 2, n, h[n-1]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     If[i > n, 0, Expand[x*b[n-i, h[Min[n-i, i]]]]] + b[n, h[i-1]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Max[0, Exponent[#, x]]}]&[b[n, h[n]]];
Table[T[n], {n, 0, 32}] // Flatten (* Jean-François Alcover, Aug 19 2021, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A001358(j)).

Sum_{k>0} k * T(n,k) = A281617(n).

A344245 Number of partitions of n into 5 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 4, 3, 4, 4, 6, 7, 7, 6, 8, 9, 13, 11, 11, 12, 15, 16, 18, 18, 19, 19, 23, 26, 28, 27, 27, 32, 36, 37, 39, 42, 45, 44, 51, 55, 58, 55, 57, 66, 71, 75, 76, 82, 84, 87, 93, 104, 103, 103, 105, 119, 131, 130, 134, 141, 145, 151, 163, 173, 176, 173

Offset: 20

Wesley Ivan Hurt, May 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 20..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358 (semiprimes).
Cf. A340756.
Column k=5 of A344447.

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} [Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^5] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

cp's OEIS Frontend

A344447 Number T(n,k) of partitions of n into k semiprimes; triangle T(n,k), n>=0, read by rows.

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