A114976 -id:A114976 - cp's OEIS frontend

A051293 Number of nonempty subsets of {1,2,3,...,n} whose elements have an integer average.

1, 2, 5, 8, 15, 26, 45, 76, 135, 238, 425, 768, 1399, 2570, 4761, 8856, 16567, 31138, 58733, 111164, 211043, 401694, 766417, 1465488, 2807671, 5388782, 10359849, 19946832, 38459623, 74251094, 143524761, 277742488, 538043663, 1043333934, 2025040765, 3933915348

Offset: 1

John W. Layman, Oct 30 1999

a(n) is asymptotic to 2^(n+1)/n. More precisely, I conjecture for any m > 0, a(n) = 2^(n+1)/n * Sum_{k=0..m} A000670(k)/n^k + o(1/n^(m+1)) (A000670 = preferential arrangements of n labeled elements) which can be written a(n) = 2^n/n * 2 + Sum_{k=1..m} A000629(k)/n^k + o(1/n^(m+1)) (A000629 = necklaces of sets of labeled beads). In fact I conjecture that a(n) = 2^(n+1)/n * (1 + 1/n + 3/n^2 + 13/n^3 + 75/n^4 + 541/n^5 + o(1/n^5)). - Benoit Cloitre, Oct 20 2002

A082550(n) = a(n+1) - a(n). - Reinhard Zumkeller, Feb 19 2006

			a(4) = 8 because each of the 8 subsets {1}, {2}, {3}, {4}, {1,3}, {2,4}, {1,2,3}, {2,3,4} has an integer average.

Alois P. Heinz, Table of n, a(n) for n = 1..3332 (first 300 terms from T. D. Noe)
63rd Annual William Lowell Putnam Mathematical Competition, Problem A3, Mathematics Magazine 76 (2003), 76-80.

Row sums of A061865 and A327481.

Cf. A000629, A000670, A082550, A114976.

with(numtheory):
b:= n-> add(2^(n/d)*phi(d), d=select(x-> x::odd, divisors(n)))/n:
a:= proc(n) option remember; `if`(n<1, 0, b(n)-1+a(n-1)) end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 15 2019

Table[ Sum[a = Select[Divisors[i], OddQ[ # ] & ]; Apply[Plus, 2^(i/a)*EulerPhi[a]]/i, {i, n}] - n, {n, 34}]
Table[Count[Subsets[Range[n]],?(IntegerQ[Mean[#]]&)],{n,35}] (* _Harvey P. Dale, Apr 14 2018 *)

a(n)=sum(k=1,n,sumdiv(k,d,d%2*2^(k/d)*eulerphi(d))/k-1)

from sympy import totient, divisors
def A051293(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))-n # Chai Wah Wu, Feb 22 2023

a(n) = Sum_{i=1..n} (A063776(i) - 1).

Extended by Robert G. Wilson v, Oct 16 2002

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

Original entry on oeis.org

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

Ivan N. Ianakiev, Jul 15 2019

nonn

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

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

Cf. A051293, A114976, A127542.

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

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

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

a(n) < A051293(n).

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

cp's OEIS Frontend

A051293 Number of nonempty subsets of {1,2,3,...,n} whose elements have an integer average.

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