A344257 -id:A344257 - 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).

A344255 Number of partitions of n into 8 semiprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 4, 3, 5, 3, 5, 5, 8, 8, 9, 8, 12, 12, 17, 16, 18, 18, 22, 25, 30, 29, 33, 36, 44, 45, 51, 54, 59, 63, 71, 78, 87, 90, 99, 106, 120, 124, 136, 147, 157, 166, 182, 199, 216, 223, 238, 259, 280, 298, 314

Offset: 32

Wesley Ivan Hurt, May 12 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 32..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=8 of A344447.

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, 9}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 8}];
Table[a[n], {n, 32, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz in A344257 *)

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} [Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.

a(n) = [x^n y^8] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

A344256 Number of partitions of n into 9 semiprime parts.

Original entry on oeis.org

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

Wesley Ivan Hurt, May 13 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 36..10000
Index entries for sequences related to partitions

Cf. A001222 (Omega), A001358.

Column k=9 of A344447.

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 *)

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

More terms from Alois P. Heinz, May 18 2021

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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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A344255 Number of partitions of n into 8 semiprime parts.

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A344256 Number of partitions of n into 9 semiprime parts.

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