A299195 -id:A299195 - cp's OEIS frontend

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

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

A347592 Number of compositions (ordered partitions) of n^4 into at most n fourth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 31, 7, 31, 1, 1417, 391, 158341, 7, 15873901, 14001, 842422057, 579379362, 56336866681, 46619148182102, 778545853570143, 178899570114917382, 3463518647447701366, 503865540593731411886, 11759574310854845083219, 1403272743542390575650072

Offset: 0

Ilya Gutkovskiy, Sep 08 2021

nonn

Cf. A000583, A299195, A307739, A346566, A347590, A347591.

a(19)-a(24) from Alois P. Heinz, Sep 08 2021

cp's OEIS Frontend

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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A347592 Number of compositions (ordered partitions) of n^4 into at most n fourth powers.

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