A240089 -id:A240089 - cp's OEIS frontend

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

Ilya Gutkovskiy, Dec 05 2020

nonn

			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}.

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

Cf. A051293, A240089, A326027, A339453.

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

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

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

A240090 Number of partitions of n that have integer contraharmonic mean.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 3, 7, 5, 8, 5, 17, 8, 21, 14, 31, 18, 49, 28, 56, 42, 90, 52, 146, 77, 189, 118, 257, 158, 370, 219, 530, 313, 724, 412, 999, 578, 1372, 809, 1837, 1094, 2515, 1472, 3387, 1948, 4584, 2656, 6145, 3527, 8114, 4665, 10784, 6225, 14196, 8150

Offset: 1

Clark Kimberling and Peter J. C. Moses, Apr 01 2014

The contraharmonic mean of a set {x(1),..,x(k)} is defined as (x(1)^2 + ... + x(k)^2)/(x(1) + ... + x(k)); if the set is a partition of n, this mean is (x(1)^2 + ... + x(k)^2)/n, which is the square of the root mean square of the partition, discussed at A240090.

			a(10) counts these 8 partitions: [10], [6,1,1,1,1], [5,5], [5,1,1,1,1,1], [4,3,2,1], [3,2,2,1,1,1], [2,2,2,2,2], [1,1,1,1,1,1,1,1,1,1]; e.g., [4,3,2,1] has contraharmonic mean (16 + 9 + 4 + 1)/10 = 3.

Cf. A240089.

z = 15; ColumnForm[t = Map[Select[IntegerPartitions[#],      IntegerQ[RootMeanSquare[#]] &] &, Range[z]]] (* shows the partitions *)
t1 = Map[Length, t]  (* A240089 *)
ColumnForm[u = Map[Select[IntegerPartitions[#], IntegerQ[ContraharmonicMean[#]] &] &, Range[z]]] (* shows the partitions *)
t2 = Map[Length, u]  (* A240090 *)

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

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