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A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.

%I A357413 #33 Mar 07 2025 07:49:41
%S A357413 0,1,1,2,2,3,3,4,4,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,19,19,24,
%T A357413 24,25,25,26,26,27,27,28,28,29,29,30,30,31,31,32,32,39,39,40,40,49,49,
%U A357413 50,50,51,51,52,52,53,53,54,54,55,55,62,62,63,63,64,64,65,65,66,66,67,67,90,90,91,91,92,92
%N A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.
%C A357413 The geometric mean of a subset such as in name must be an odd number in {1..n} which might ease the search for terms. - _David A. Corneth_, Sep 29 2022
%H A357413 Max Alekseyev, <a href="/A357413/b357413.txt">Table of n, a(n) for n = 0..256</a>
%F A357413 a(2*n-1) = a(2*n) for n >= 1. - _David A. Corneth_, Sep 29 2022
%F A357413 a(n) = A326027(n) - A357414(n). - _Max Alekseyev_, Mar 01 2025
%e A357413 a(9) = 7 subsets: {1}, {3}, {5}, {7}, {9}, {1, 9} and {1, 3, 9}.
%o A357413 (Python)
%o A357413 from functools import lru_cache
%o A357413 from sympy import integer_nthroot
%o A357413 def cond(p, c): r, b = integer_nthroot(p, c); return b and r&1
%o A357413 @lru_cache(maxsize=None)
%o A357413 def b(n, p, c):
%o A357413     if n == 0: return int (c > 0 and cond(p, c))
%o A357413     return b(n-1, p, c) + b(n-1, p*n, c+1) if n&1 else b(n-1, p, c)
%o A357413 @lru_cache(maxsize=None)
%o A357413 def a(n): return b(n, 1, 0) if n&1 else b(n-1, 1, 0) if n else 0
%o A357413 print([a(n) for n in range(41)]) # _Michael S. Branicky_, Sep 29 2022
%Y A357413 Cf. A001055, A326027, A357355, A357411, A357414, A357415.
%K A357413 nonn
%O A357413 0,4
%A A357413 _Ilya Gutkovskiy_, Sep 27 2022
%E A357413 a(24)-a(34) from _Michael S. Branicky_, Sep 29 2022
%E A357413 a(35)-a(70) from _David A. Corneth_, Sep 29 2022
%E A357413 a(0) prepended and terms a(71) onward added by _Max Alekseyev_, Mar 06 2025