A259201 -id:A259201 - cp's OEIS frontend

A341460 Number of partitions of n into 10 nonprime parts.

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 10, 12, 15, 17, 20, 24, 28, 32, 38, 44, 51, 60, 68, 79, 92, 104, 122, 139, 157, 181, 208, 234, 270, 304, 347, 391, 445, 499, 569, 636, 724, 805, 913, 1015, 1150, 1274, 1440, 1592, 1796, 1980, 2231, 2455

Offset: 10

Ilya Gutkovskiy, Feb 12 2021

nonn

Cf. A002095, A005171, A018252, A062610, A259201, A341408, A341451, A341452, A341453, A341454, A341455, A341457.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(isprime(i), 0, b(n-i, min(n-i, i), t-1))))
    end:
a:= n-> b(n$2, 10):
seq(a(n), n=10..69);  # Alois P. Heinz, Feb 12 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 10];
Table[a[n], {n, 10, 69}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{10}],?(NoneTrue[#,PrimeQ]&)],{n,10,70}] (* _Harvey P. Dale, Sep 01 2024 *)

A309439 Number of prime parts in the partitions of n into 10 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 11, 17, 30, 45, 72, 104, 157, 210, 298, 396, 537, 698, 924, 1176, 1521, 1909, 2418, 2991, 3729, 4560, 5610, 6795, 8254, 9906, 11919, 14180, 16908, 19972, 23615, 27706, 32527, 37917, 44227, 51267, 59425, 68525, 79007

Offset: 0

Wesley Ivan Hurt, Aug 03 2019

nonn

Index entries for sequences related to partitions

Cf. A010051, A259201.

Table[Count[Flatten[IntegerPartitions[n,{10}]],?PrimeQ],{n,0,50}] (* _Harvey P. Dale, Dec 26 2020 *)

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} (c(r) + c(q) + c(p) + c(o) + c(m) + c(l) + c(k) + c(j) + c(i) + c(n-i-j-k-l-m-o-p-q-r)), where c = A010051.

cp's OEIS Frontend

A341460 Number of partitions of n into 10 nonprime parts.

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