A025401 -id:A025401 - cp's OEIS frontend

A374257 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

-1, 325, 5104, 16578, 70211956, 201968338, 1690592199245

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(2) = 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A025304, A025401, A146756, A374227, A374256.

from itertools import count
from sympy.solvers.diophantine.diophantine import power_representation
def A374257(n): return next(m for m in count(1) if len(list(power_representation(m,n,n)))==3) if n>1 else -1 # Chai Wah Wu, Jul 01 2024

a(7) from Michael S. Branicky, Jul 09 2024

A350270 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

1, 1729, 5104, 4445, 4509, 4662, 5454, 6210, 9045, 11124, 14967, 17964, 22051, 26209, 32697, 39564, 46908, 56070, 66222, 78912, 92961, 105841, 123732, 143200, 162801, 188154, 212220, 241614, 271405, 307448, 344016, 383607, 428624, 475273, 529830, 586664, 645120

Offset: 1

Ilya Gutkovskiy, Dec 22 2021

nonn

			For n = 2: 1729 = 1^3 + 12^3 = 9^3 + 10^3.
For n = 3: 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Index entries for sequences related to sums of cubes

Cf. A000578, A025401, A025419, A025421, A275154, A350241.

a(16)-a(27) from Michael S. Branicky, Dec 22 2021

More terms from Jinyuan Wang, Dec 30 2021

A350430 a(n) is the smallest n-th power which can be represented as the sum of n distinct positive n-th powers in exactly n ways, or -1 if none exists.

Original entry on oeis.org

1, 625, 157464

Offset: 1

Ilya Gutkovskiy, Dec 30 2021

From Jon E. Schoenfield, Dec 30 2021: (Start)
222000^4 < a(4) < 4891341^4 = lcm(2829, 12259, 16359, 30381)^4 (see A039664, including the Wroblewski link).
10000^5 <= a(5) < 12528^5 = lcm(72, 1044, 1392, 2088, 3132)^5 (see A063923, including the Waldby link; note that, although the terms of A063923 include 72, 144, 1044, 1392, and 2088, whose LCM is only 4176, the primitive solution in which the sum of 5 distinct 5th powers is 144^5 is 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 144^5, which is not the sum of 5 positive n-th powers).
Conjecture: a(6) = -1. (End)

			For n = 2: 625 = 25^2 = 7^2 + 24^2 = 15^2 + 20^2.
For n = 3: 157464 = 54^3 = 6^3 + 36^3 + 48^3 = 12^3 + 19^3 + 53^3 = 27^3 + 36^3 + 45^3.

Cf. A007666, A025303, A025401, A030052, A039664, A063923, A146756, A230718, A331675.

cp's OEIS Frontend

A374257 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 3 ways, or -1 if no such number exists.

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A350270 a(n) is the smallest number which can be represented as the sum of n distinct positive cubes in exactly n ways, or 0 if no such number exists.

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A350430 a(n) is the smallest n-th power which can be represented as the sum of n distinct positive n-th powers in exactly n ways, or -1 if none exists.

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