A357416 -id:A357416 - 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

A357414 Number of nonempty subsets of {1..n} whose elements have an even geometric mean.

Original entry on oeis.org

0, 0, 1, 1, 4, 4, 5, 5, 8, 12, 13, 13, 20, 20, 21, 21, 30, 30, 59, 59, 62, 62, 63, 63, 94, 104, 105, 187, 190, 190, 191, 191, 306, 306, 307, 307, 564, 564, 565, 565, 582, 582, 583, 583, 586, 600, 601, 601, 1120, 1134, 1275, 1275, 1278, 1278, 2125, 2125, 2144, 2144, 2145, 2145, 2360, 2360, 2361, 2381, 3938, 3938, 3939, 3939, 3942, 3942, 3943, 3943, 6560, 6560, 6561, 9663, 9666

Offset: 0

Ilya Gutkovskiy, Sep 27 2022

nonn

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

Max Alekseyev, Table of n, a(n) for n = 0..255

Cf. A326027, A357356, A357412, A357413, A357416.

from functools import lru_cache
from sympy import integer_nthroot
def cond(p, c): r, b = integer_nthroot(p, c); return b and r&1 == 0
@lru_cache(maxsize=None)
def b(n, p, c):
    if n == 0: return int (c > 0 and cond(p, c))
    return b(n-1, p, c) + b(n-1, p*n, c+1)
a = lambda n: b(n, 1, 0)
print([a(n) for n in range(26)]) # Michael S. Branicky, Sep 29 2022

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

a(n) = A326027(n) - A357413(n). - Max Alekseyev, Mar 06 2025

a(24)-a(41) from Michael S. Branicky, Sep 30 2022
Terms a(42) onward from Max Alekseyev, Oct 11 2023
a(0) prepended by Max Alekseyev, Mar 06 2025

A357415 Number of nonempty subsets of {1..n} whose elements have an odd root mean square.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 7, 9, 16, 26, 41, 85, 142, 254, 461, 825, 1454, 2506, 4535, 7987, 14352, 26178, 47861, 87945, 162486, 304864, 565217, 1064529, 1992628, 3742934, 7034489, 13214869, 24924676, 46926388, 88812537, 167903969, 318619708, 604909434, 1150800393

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

nonn

			a(10) = 9 subsets: {1}, {3}, {5}, {7}, {9}, {1, 7}, {1, 5, 7}, {2, 3, 6, 8, 9, 10} and {2, 3, 6, 7, 8, 9, 10}.

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A339454, A357355, A357411, A357413, A357416.

a(n) = A339454(n) - A357416(n).

a(24)-a(41) from Alois P. Heinz, Sep 27 2022

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

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

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

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

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

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

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

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

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A357414 Number of nonempty subsets of {1..n} whose elements have an even geometric mean.

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A357415 Number of nonempty subsets of {1..n} whose elements have an odd root mean square.

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Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Formula

Extensions