A071330 -id:A071330 - cp's OEIS frontend

A341131 Number of partitions of n into 10 prime powers (including 1).

1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 41, 56, 67, 87, 105, 134, 156, 196, 228, 278, 320, 387, 439, 526, 593, 698, 783, 915, 1014, 1179, 1304, 1497, 1648, 1884, 2058, 2342, 2551, 2882, 3130, 3524, 3802, 4266, 4595, 5125, 5504, 6124, 6548, 7263, 7750, 8558

Offset: 10

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A341112, A341122, A341123, A341124, A341125, A341126, A341127.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
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`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 10):
seq(a(n), n=10..60);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
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[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 10];
Table[a[n], {n, 10, 60}] (* Jean-François Alcover, Feb 27 2022, after Alois P. Heinz *)

A290135 Numbers that are the sum of two proper prime powers (A246547).

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 24, 25, 29, 31, 32, 33, 34, 35, 36, 40, 41, 43, 48, 50, 52, 53, 54, 57, 58, 59, 64, 65, 68, 72, 73, 74, 76, 80, 81, 85, 89, 90, 91, 96, 97, 98, 106, 108, 113, 125, 128, 129, 130, 132, 133, 134, 136, 137, 141, 144, 145, 146, 148, 150, 152, 153, 155, 157, 160, 162, 170, 173, 174, 177, 178

Offset: 1

Ilya Gutkovskiy, Jul 20 2017

nonn

Is 2213 the largest prime term that can be expressed as the sum of two proper prime powers in more than one way? - Altug Alkan, Jul 22 2017

			13 is in the sequence because 13 = 2^2 + 3^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014091, A070049, A071330, A071331, A225102, A225103, A246547.

N:= 1000: # to get all terms <= N
P:= select(isprime, [$2..floor(sqrt(N))]):
PP:= {seq(seq(p^j, j=2..floor(log[p](N))),p=P)}:
A:= select(`<=`,{seq(seq(PP[i]+PP[j],j=1..i),i=1..nops(PP))},N):
sort(convert(A,list)); # Robert Israel, Jul 21 2017

nmax = 180; f[x_] := Sum[Boole[PrimePowerQ[k] && PrimeOmega[k] > 1] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

Exponents in expansion of (Sum_{k>=1} x^A246547(k))^2.

cp's OEIS Frontend

A341131 Number of partitions of n into 10 prime powers (including 1).

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A290135 Numbers that are the sum of two proper prime powers (A246547).

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