A299169 -id:A299169 - cp's OEIS frontend

A299195 Number of ordered ways of writing n^4 as a sum of n fourth powers of positive integers.

1, 1, 0, 0, 0, 30, 6, 0, 0, 0, 360, 157080, 0, 12586860, 0, 714233520, 579379361, 48062263014, 46026944529624, 759085890469938, 170947379002578290, 3331302954541376850, 479526242126281889924, 11322897238957194004884, 1341983461418984670506352, 31585668052999315295625900

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

			a(6) = 6 because we have [256, 256, 256, 256, 256, 16], [256, 256, 256, 256, 16, 256], [256, 256, 256, 16, 256, 256], [256, 256, 16, 256, 256, 256], [256, 16, 256, 256, 256, 256] and [16, 256, 256, 256, 256, 256].

Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A046042, A259793, A298330, A298672, A298859, A299169.

a[0] = 1; a[n_] := Coefficient[Sum[x^k^4, {k, n-1}]^n // Expand, x, n^4]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 25}] (* Jean-François Alcover, Feb 05 2018 *)

a(n) = [x^(n^4)] (Sum_{k>=1} x^(k^4))^n.

More terms from Alois P. Heinz, Feb 04 2018

A307644 Number of partitions of n^4 into exactly n nonzero fourth powers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 6, 0, 27, 13, 59, 390, 661, 4933, 9760, 49415, 101967, 341887, 702884, 2209559, 5361004, 15472531, 34165997, 82258594, 193682533, 490404772, 1210929426, 2725005202, 6283337761, 13672859806, 34906926846

Offset: 0

Ilya Gutkovskiy, Apr 19 2019

nonn

			11^4 =
1^4 + 2^4 + 2^4 + 4^4 + 4^4 + 4^4 + 4^4 + 6^4 + 8^4 + 8^4 + 8^4 =
2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 2^4 + 6^4 + 6^4 + 6^4 + 8^4 + 9^4 =
2^4 + 2^4 + 2^4 + 4^4 + 6^4 + 6^4 + 6^4 + 6^4 + 6^4 + 6^4 + 9^4,
so a(11) = 3.

Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A259793, A299169, A299195, A307643, A319435.

a(20)-a(28) from Vaclav Kotesovec, Apr 20 2019

a(29)-a(37) from Vaclav Kotesovec, Apr 23 2019

A307739 Number of partitions of n^4 into at most n fourth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 5, 3, 5, 2, 27, 4, 78, 14, 152, 551, 1331, 7377, 15504, 87583, 190028, 768864, 1510305, 5413291, 12221733

Offset: 0

Ilya Gutkovskiy, Apr 25 2019

			10^4 =
4^4 + 4^4 + 6^4 + 8^4 + 8^4 =
2^4 + 4^4 + 4^4 + 4^4 + 4^4 + 4^4 + 4^4 + 4^4 + 8^4 + 8^4,
so a(10) = 3.

Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A105152, A259793, A299169, A299195, A307644, A307738.

from sympy.solvers.diophantine.diophantine import power_representation
def a(n):
    if n < 2: return 1
    return sum(len(list(power_representation(n**4, 4, j))) for j in range(1, n+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jul 09 2024

a(21)-a(27) from Michael S. Branicky, Jul 09 2024

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A299195 Number of ordered ways of writing n^4 as a sum of n fourth powers of positive integers.

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A307644 Number of partitions of n^4 into exactly n nonzero fourth powers.

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A307739 Number of partitions of n^4 into at most n fourth powers.

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