A357412 Number of nonempty subsets of {1..n} whose elements have an even harmonic mean.

0, 1, 1, 2, 2, 7, 7, 8, 8, 9, 9, 16, 16, 17, 27, 28, 28, 55, 55, 106, 110, 111, 111, 216, 216, 217, 217, 634, 634, 1155, 1155, 1156, 2286, 2287, 3749, 5840, 5840, 5841, 5847, 11834, 11834, 23409, 23409, 23470, 47202, 47203, 47203, 94762, 94762, 94763, 94763, 195092

Offset

1,4

Links

Martin Fuller, Table of n, a(n) for n = 1..75

Formula

a(p) = a(p-1) for prime p > 2. - Michael S. Branicky, Sep 30 2022

Example

a(11) = 9 subsets: {2}, {4}, {6}, {8}, {10}, {3, 6}, {1, 3, 6}, {3, 4, 6} and {1, 2, 3, 6}.

Prog

(Python)
from fractions import Fraction
from functools import lru_cache
def cond(s, c): h = c/s; return h.denominator == 1 and h.numerator&1 == 0
@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+Fraction(1, n), c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Sep 29 2022
(PARI) gppf(n)=if(abs(n)>1, vecmax(apply(pe->pe[1]^pe[2], Col(factor(n)))), n);
N(s, k, v, m)={
  /* number of ways to sum to s using k distinct reciprocals from v[1..m] */
  /* v must be in ascending order of greatest prime-power factor gppf() */
  if(k==1, return(if(numerator(s)==1, sum(i=1, m, v[i]==denominator(s)))));
  my(r=0, f=gppf(denominator(s)));
  forstep(i=m, k, -1, if(gppf(v[i]) < f, break); if(s > 1/v[i], r += N(s - 1/v[i], k-1, v, i-1)));
  r};
a(n)=my(v=vecsort([1..n], gppf)); n\2 + sum(k=2, n, sum(j=2, n-1, if(j%2==0, N(k/j, k, v, n))));
\\ Martin Fuller, Apr 24 2026

Crossrefs

Cf. A339453, A357356, A357411, A357414, A357416.

Sequence in context: A064288 A394090 A054085 * A021443 A045923 A306238

Adjacent sequences: A357409 A357410 A357411 * A357413 A357414 A357415

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 27 2022

Extensions

a(24)-a(35) from Michael S. Branicky, Sep 30 2022

a(36) onward from Martin Fuller, Apr 24 2026

Status

approved