A309160 Number of nonempty subsets of [n] whose elements have a prime average.

0, 1, 4, 6, 11, 15, 22, 40, 72, 118, 199, 355, 604, 920, 1306, 1906, 3281, 6985, 16446, 38034, 82490, 168076, 325935, 604213, 1068941, 1815745, 3038319, 5246725, 9796132, 19966752, 42918987, 92984247, 197027405, 402932711, 792381923, 1499918753, 2746078246

Offset: 1

Author

Ivan N. Ianakiev, Jul 15 2019

Keywords

• nonn

Examples

			a(3) = 4 because 4 of the subsets of [3], namely {2}, {3}, {1,3}, {1,2,3}, have prime averages.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A051293, A114976, A127542.

Programs

Maple

b:= proc(n, s, c) option remember; `if`(n=0,
      `if`(c>0 and denom(s)=1 and isprime(s), 1, 0),
       b(n-1, s, c)+b(n-1, (s*c+n)/(c+1), c+1))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 15 2019

Mathematica

a[n_]:=Length[Select[Subsets[Range[n]],PrimeQ[Mean[#]]&]]; a/@Range[25]

Python

from sympy import isprime
from functools import lru_cache
def cond(s, c): q, r = divmod(s, c); return r == 0 and isprime(q)
@lru_cache(maxsize=None)
def b(n, s, c):
    if n == 0: return int (c > 0 and cond(s, c))
    return b(n-1, s, c) + b(n-1, s+n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Sep 25 2022

Formula

a(n) < A051293(n).

Extensions

a(26)-a(37) from Alois P. Heinz, Jul 15 2019