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A339485
Number of subsets of the first n primes whose elements have a prime average.
2
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
1,2
LINKS
Eric W. Weisstein's World of Mathematics, Arithmetic Mean
EXAMPLE
a(5) = 9 subsets: {2}, {3}, {5}, {7}, {11}, {3, 7}, {3, 11}, {3, 5, 7} and {3, 7, 11}.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 06 2020
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
STATUS
approved