A299168 -id:A299168 - cp's OEIS frontend

A341459 Number of compositions of n^2 into n prime parts.

1, 0, 1, 4, 22, 241, 2696, 35218, 529888, 8998419, 169486964, 3496417024, 78344008779, 1891733424205, 48923563968087, 1347813311456319, 39371345548420060, 1214570579814316742, 39430967625404799740, 1343040950675651131103, 47862610677098010505554

Offset: 0

Alois P. Heinz, Feb 12 2021

nonn

			a(3) = 4: 333, 225, 252, 522.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A023360, A035026, A121303, A299168.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(isprime(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n^2, n):
seq(a(n), n=0..22);

b[n_, t_] := b[n, t] =
     If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[
     If[PrimeQ[j], b[n-j, t-1], 0], {j, 1, n}]]];
a[n_] := b[n^2, n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

a(n) = A121303(n^2,n).

A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 30, 77, 252, 682, 2136, 6182, 18766, 56173, 169351, 512990, 1551828, 4720170, 14348289, 43751984, 133502873, 408029510, 1248460587, 3823949824, 11724787763, 35980251181, 110510334780, 339674840715, 1044812449722, 3215861978150, 9904301974294, 30521063942312, 94103983534015

Offset: 0

Ilya Gutkovskiy, Mar 29 2018

nonn

Number of partitions of n into prime parts of n kinds.

Index entries for sequences related to partitions

Cf. A000607, A008485, A030009, A117278, A219224, A259254, A291647, A298436, A299168.

Table[SeriesCoefficient[Product[1/(1 - x^Prime[k])^n, {k, 1, n}], {x, 0, n}], {n, 0, 31}]

A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 25, 9, 61, 91, 99, 151, 901, 303, 1759, 3379, 5239, 4713, 8227, 12901, 12537, 23059, 65239, 159421, 232369, 489817, 351237, 726295, 564363, 1101883, 2517865, 6916027, 11825821, 4942227, 27166753, 21280053, 39547957, 52630273, 113638975

Offset: 1

Ilya Gutkovskiy, Jan 31 2020

nonn

			a(4) = 3 because we have [7], [5, 2] and [2, 5].

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to compositions

Cf. A000040, A023360, A056768, A070215, A219107, A265112, A299168.

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(ithprime(n), n, 0):
seq(a(n), n=1..42);  # Alois P. Heinz, Jan 31 2020

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, Function[p, If[p > n, 0, b[n - p, i - 1, t + 1]]][Prime[i]] + b[n, i - 1, t]]];
a[n_] := b[Prime[n], n, 0];
Array[a, 42] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

a(n) = A219107(A000040(n)).

cp's OEIS Frontend

A341459 Number of compositions of n^2 into n prime parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula