A339409 -id:A339409 - cp's OEIS frontend

A339408 Number of compositions (ordered partitions) of n into an even number of primes.

1, 0, 0, 0, 1, 2, 1, 2, 3, 6, 9, 8, 16, 24, 40, 52, 72, 112, 172, 256, 364, 528, 804, 1188, 1757, 2548, 3782, 5614, 8308, 12214, 17979, 26586, 39352, 58044, 85608, 126248, 186630, 275556, 406737, 600066, 885952, 1308250, 1931473, 2850692, 4207952, 6212110, 9171800, 13538980

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(8) = 3 because we have [5, 3], [3, 5] and [2, 2, 2, 2].

Index entries for sequences related to compositions

Cf. A000040, A023360, A034008, A184198, A339380, A339409.

b:= proc(n, t) option remember; `if`(n=0, t, add(
      b(n-ithprime(j), 1-t), j=1..numtheory[pi](n)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..55);  # Alois P. Heinz, Dec 03 2020

nmax = 47; CoefficientList[Series[(1/2) (1/(1 - Sum[x^Prime[k], {k, 1, nmax}]) + 1/(1 + Sum[x^Prime[k], {k, 1, nmax}])), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^prime(k)) + 1 / (1 + Sum_{k>=1} x^prime(k))).

A339433 Number of compositions (ordered partitions) of n into an odd number of distinct primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 6, 1, 6, 1, 6, 6, 6, 1, 12, 7, 12, 12, 12, 13, 12, 12, 18, 18, 132, 19, 132, 25, 18, 24, 252, 30, 258, 31, 264, 156, 486, 37, 498, 157, 504, 276, 738, 175, 738, 288, 750, 528, 984, 307, 1218, 666, 1110, 780, 6378, 679, 6618, 799, 1716, 1272

Offset: 0

Ilya Gutkovskiy, Dec 04 2020

nonn

			a(10) = 6 because we have [5, 3, 2], [5, 2, 3], [3, 5, 2], [3, 2, 5], [2, 5, 3] and [2, 3, 5].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A000040, A184172, A219107, A332304, A339383, A339409, A339432.

b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!, (s->
     `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(ithprime(i)))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 04 2020

b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, Function[s, If[s > n, 0, b[n, i + 1, p] + b[n - s, i + 1, p + 1]]][Prime[i]]];
a[n_] := b[n, 1, 0];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A339408 Number of compositions (ordered partitions) of n into an even number of primes.

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