A339485 Number of subsets of the first n primes whose elements have a prime average.

1, 2, 3, 6, 9, 12, 17, 30, 51, 88, 149, 264, 439, 746, 1261, 2234, 4211, 7996, 14899, 28048, 54037, 106442, 208625, 398588, 735365, 1331590, 2421573, 4481896, 8504953, 16497150, 32595915, 64614636, 127968263, 252470776, 495388085, 962475122, 1847742473, 3504948056

Offset: 1

Author

Ilya Gutkovskiy, Dec 06 2020

Keywords

• nonn

Examples

			a(5) = 9 subsets: {2}, {3}, {5}, {7}, {11}, {3, 7}, {3, 11}, {3, 5, 7} and {3, 7, 11}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..150
• Eric W. Weisstein's World of Mathematics, Arithmetic Mean

Crossrefs

Cf. A000040, A051293, A071810, A127542, A309160.

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+ithprime(n))/(c+1), c+1))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 08 2020

Mathematica

b[n_, s_, c_] := b[n, s, c] = If[n == 0,
     If[c > 0 && Denominator[s] == 1 && PrimeQ[s], 1, 0],
     b[n-1, s, c] + b[n-1, (s*c + Prime[n])/(c+1), c+1]];
a[n_] := b[n, 0, 0];
Array[a, 40] (* Jean-François Alcover, Jul 09 2021, after Alois P. Heinz *)

Python

from sympy import prime, isprime
from itertools import chain, combinations
def powerset(s): # skip empty set and singletons
    return chain.from_iterable(combinations(s, r) for r in range(2,len(s)+1))
def a(n):
    out = n  # count all singletons
    for s in powerset([prime(i) for i in range(1, n+1)]):
        ss = sum(s)
        if ss%len(s) == 0:
            if isprime(ss//len(s)): out += 1
    return out
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Dec 06 2020

Python

from itertools import combinations
from sympy import prime
def A339485(n):
    c, primeset2 = n, set(prime(i) for i in range(1,n))
    primeset = primeset2 | {prime(n)}
    for l in range(2,n+1):
        for d in combinations(primeset,l):
            a, b = divmod(sum(d),l)
            if b == 0 and a in primeset2:
                c += 1
    return c # Chai Wah Wu, Dec 07 2020

Python

from functools import lru_cache
from sympy import sieve, isprime
@lru_cache(maxsize=None)
def b(n, s, c):
    if n == 0: return int(c and s%c == 0 and isprime(s//c))
    return b(n-1, s, c) + b(n-1, s+sieve[n], c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Oct 06 2022

Extensions

a(10)-a(30) from Michael S. Branicky, Dec 06 2020

a(31)-a(34) from Chai Wah Wu, Dec 07 2020

a(35)-a(36) from Michael S. Branicky, Dec 08 2020

a(37)-a(38) from Chai Wah Wu, Dec 08 2020