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Number of subsets of {1..n} whose root mean square is an integer.
7

%I #10 Oct 06 2022 19:00:06

%S 1,2,3,4,5,6,9,10,15,20,29,52,87,166,311,538,943,1682,2915,5054,8905,

%T 15904,28533,51826,95191,175402,325777,607542,1134191,2128922,3986433,

%U 7485522,14065135,26446388,49796025,93920770,177470237,335780796,636883269,1209603646

%N Number of subsets of {1..n} whose root mean square is an integer.

%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Root-Mean-Square.html">Root-Mean-Square</a>

%e a(9) = 15 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {1, 7}, {1, 5, 7}, {1, 3, 5, 8, 9}, {3, 4, 5, 7, 9}, {1, 3, 5, 6, 8, 9} and {3, 4, 5, 6, 7, 9}.

%o (Python)

%o from functools import lru_cache

%o from sympy.ntheory.primetest import is_square

%o def cond(sos, c): return c > 0 and sos%c == 0 and is_square(sos//c)

%o @lru_cache(maxsize=None)

%o def b(n, sos, c):

%o if n == 0: return int(cond(sos, c))

%o return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)

%o a = lambda n: b(n, 0, 0)

%o print([a(n) for n in range(1, 41)]) # _Michael S. Branicky_, Oct 06 2022

%Y Cf. A051293, A240089, A326027, A339453.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Dec 05 2020

%E a(23)-a(40) from _Alois P. Heinz_, Dec 05 2020