A374256 -id:A374256 - 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

A374270 a(n) is the smallest number which can be represented as the sum of n distinct n-th powers of primes in exactly 2 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 410, 185527, 6539044, 2250298051, 1397942436813, 56190747296833091

Offset: 1

Ilya Gutkovskiy, Jul 02 2024

			a(2) = 410 = 7^2 + 19^2 = 11^2 + 17^2.
a(3) = 185527 = 13^3 + 43^3 + 47^3 = 19^3 + 31^3 + 53^3.

Cf. A349764, A374256, A374271, A374272.

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

A374269 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 4 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 1105, 13896, 300834, 1386406515, 2063792939

Offset: 1

Ilya Gutkovskiy, Jul 02 2024

			a(2) = 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.
a(3) = 13896 = 1^3 + 12^3 + 23^3 = 2^3 + 4^3 + 24^3 = 4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3.

Cf. A025305, A146756, A374228, A374256, A374257.

a(6) from Michael S. Branicky, Jul 09 2024

A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

Original entry on oeis.org

19, 90, 162, 299, 509, 816, 1248, 1837, 2619, 3634, 4926, 6543, 8537, 10964, 13884, 17361, 21463, 26262, 31834, 38259, 45621, 54008, 63512, 74229, 86259, 99706, 114678, 131287, 149649, 169884, 192116, 216473, 243087, 272094, 303634, 337851, 374893, 414912, 458064, 504509

Offset: 3

Ilya Gutkovskiy, Nov 05 2024

nonn

From David A. Corneth, Nov 06 2024: (Start)
a(n) <= (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. Proof:
A polygonal number is of the form P(m, n) = m/2 * ((n - 2) * m - n + 4).
We have P(n - 5, n) + P(n - 4, n) + P(n, n) = P(n - 6, n) + P(n - 2, n) + P(n - 1, n) = (3*n^3 - 18*n^2 + 21*n) / 2.
This lets us find the upper bound on a(n) by making two lists from 1 through n + 3. From one of them we remove n-2, n-1 and n + 3 and from the other we remove n-3, n+1 and n+2. The sum for remaining polygonal numbers is the same giving an upper bound on a(n) which turns out to be (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 (End)

			a(3) = 19 = 1 + 3 + 15 = 3 + 6 + 10.
a(4) = 90 = 1^2 + 2^2 + 6^2 + 7^2 = 1^2 + 3^2 + 4^2 + 8^2.

Eric Weisstein's World of Mathematics, Polygonal Number

Cf. A006484, A025377, A057145, A350405, A374144, A374256, A374287.

From David A. Corneth, Nov 06 2024: (Start)
a(n) >= A006484(n).
Conjecture: a(n) = (n^4 - 2*n^3 + 38*n^2 - 85*n + 72)/6 for n >= 5. (End)
Conjectured g.f.: x^3*(19 - 5*x - 98*x^2 + 199*x^3 - 171*x^4 + 72*x^5 - 12*x^6) / (1 - x)^5.

a(12)-a(36) from Michael S. Branicky, Nov 06 2024

More terms from David A. Corneth, Nov 10 2024

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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A374270 a(n) is the smallest number which can be represented as the sum of n distinct n-th powers of primes in exactly 2 ways, or -1 if no such number exists.

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A374269 a(n) is the smallest number which can be represented as the sum of n distinct positive n-th powers in exactly 4 ways, or -1 if no such number exists.

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A377729 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly 2 ways.

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