A339454 Number of subsets of {1..n} whose root mean square is an integer.

1, 2, 3, 4, 5, 6, 9, 10, 15, 20, 29, 52, 87, 166, 311, 538, 943, 1682, 2915, 5054, 8905, 15904, 28533, 51826, 95191, 175402, 325777, 607542, 1134191, 2128922, 3986433, 7485522, 14065135, 26446388, 49796025, 93920770, 177470237, 335780796, 636883269, 1209603646

Offset: 1

Author

Ilya Gutkovskiy, Dec 05 2020

Keywords

• nonn

Examples

			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}.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100
• Eric W. Weisstein's World of Mathematics, Root-Mean-Square

Crossrefs

Cf. A051293, A240089, A326027, A339453.

Programs

Python

from functools import lru_cache
from sympy.ntheory.primetest import is_square
def cond(sos, c): return c > 0 and sos%c == 0 and is_square(sos//c)
@lru_cache(maxsize=None)
def b(n, sos, c):
    if n == 0: return int(cond(sos, c))
    return b(n-1, sos, c) + b(n-1, sos+n*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

Formula

a(n) = A357415(n) + A357416(n). - Max Alekseyev, Mar 25 2025

Extensions

a(23)-a(40) from Alois P. Heinz, Dec 05 2020