A357355 -id:A357355 - cp's OEIS frontend

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

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, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 39, 39, 40, 40, 49, 49, 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

Offset: 0

Ilya Gutkovskiy, Sep 27 2022

nonn

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

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

Max Alekseyev, Table of n, a(n) for n = 0..256

Cf. A001055, A326027, A357355, A357411, A357414, A357415.

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
@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) if n&1 else b(n-1, p, c)
@lru_cache(maxsize=None)
def a(n): return b(n, 1, 0) if n&1 else b(n-1, 1, 0) if n else 0
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 29 2022

a(2*n-1) = a(2*n) for n >= 1. - David A. Corneth, Sep 29 2022

a(n) = A326027(n) - A357414(n). - Max Alekseyev, Mar 01 2025

a(24)-a(34) from Michael S. Branicky, Sep 29 2022
a(35)-a(70) from David A. Corneth, Sep 29 2022
a(0) prepended and terms a(71) onward added 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

A357356 Number of nonempty subsets of {1..n} whose elements have an even average.

Original entry on oeis.org

0, 1, 3, 4, 6, 13, 25, 38, 64, 119, 219, 384, 686, 1285, 2405, 4428, 8236, 15569, 29463, 55582, 105326, 200847, 383609, 732744, 1403004, 2694391, 5181663, 9973416, 19226166, 37125547, 71770069, 138871244, 269005540, 521666967, 1012555015, 1966957674, 3824166974

Offset: 1

Ilya Gutkovskiy, Sep 25 2022

nonn

			a(6) = 13 subsets: {2}, {4}, {6}, {1, 3}, {2, 6}, {3, 5}, {1, 2, 3}, {1, 5, 6}, {2, 4, 6}, {3, 4, 5}, {1, 4, 5, 6}, {2, 3, 5, 6} and {2, 3, 4, 5, 6}.

Cf. A051293, A309160, A357355.

from functools import lru_cache
def cond(s, c): q, r = divmod(s, c); return r == 0 and q&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+n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Sep 25 2022

a(n) = A051293(n) - A357355(n). - Michael S. Branicky, Sep 25 2022

a(24)-a(37) from Michael S. Branicky, Sep 25 2022

A357411 Number of nonempty subsets of {1..n} whose elements have an odd harmonic mean.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 6, 7, 9, 10, 10, 11, 13, 26, 26, 27, 45, 46, 74, 93, 99, 100, 162, 163, 165, 166, 458, 459, 865, 866, 866, 1647, 1669, 2724

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

			a(11) = 10 subsets: {1}, {3}, {5}, {7}, {9}, {11}, {2, 6}, {2, 3, 6}, {3, 6, 10} and {3, 5, 6, 10}.

Cf. A339453, A357355, A357412, A357413, A357415.

from fractions import Fraction
from functools import lru_cache
def cond(s, c): h = c/s; return h.denominator == 1 and h.numerator&1
@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) + 1 for prime p > 2. - Michael S. Branicky, Sep 30 2022

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

A369391 Number of nonempty subsets of {1..n} whose elements have a square average.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 20, 26, 31, 35, 41, 63, 143, 399, 1083, 2554, 5078, 8596, 12772, 17222, 21792, 27126, 36538, 62530, 146701, 412191, 1178071, 3156949, 7703823, 16992539, 33946851, 61840501, 103674983, 161604395, 236759149, 330078718, 446073360, 606337592

Offset: 1

Ilya Gutkovskiy, Jan 22 2024

nonn

			a(5) = 4 subsets: {1}, {4}, {3, 5} and {3, 4, 5}.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A051293, A126024, A309160, A339454, A357355.

a(24)-a(38) from Alois P. Heinz, Jan 22 2024

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A357413 Number of nonempty subsets of {1..n} whose elements have an odd 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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A357356 Number of nonempty subsets of {1..n} whose elements have an even average.

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A357411 Number of nonempty subsets of {1..n} whose elements have an odd harmonic mean.

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Python

Formula

Extensions

A369391 Number of nonempty subsets of {1..n} whose elements have a square average.

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Extensions