A342899 Smallest number that is the sum of five positive n-th powers in four ways.
53, 1252, 20995, 287618651, 351060139210
Offset: 2
Examples
53 = 1^2 + 1^2 + 1^2 + 1^2 + 7^2
= 1^2 + 1^2 + 1^2 + 5^2 + 5^2
= 1^2 + 2^2 + 4^2 + 4^2 + 4^2
= 1^2 + 3^2 + 3^2 + 3^2 + 5^2
= 2^2 + 2^2 + 2^2 + 4^2 + 5^2.
1252 = 1^3 + 1^3 + 5^3 + 5^3 + 10^3
= 1^3 + 2^3 + 3^3 + 6^3 + 10^3
= 3^3 + 3^3 + 7^3 + 7^3 + 8^3
= 3^3 + 4^3 + 6^3 + 6^3 + 9^3.
20995 = 1^4 + 1^4 + 1^4 + 4^4 + 12^4
= 2^4 + 3^4 + 3^4 + 3^4 + 12^4
= 2^4 + 6^4 + 9^4 + 9^4 + 9^4
= 4^4 + 6^4 + 7^4 + 7^4 + 11^4.
287618651 = 8^5 + 21^5 + 27^5 + 27^5 + 48^5
= 9^5 + 13^5 + 26^5 + 37^5 + 46^5
= 11^5 + 12^5 + 23^5 + 41^5 + 44^5
= 11^5 + 20^5 + 22^5 + 30^5 + 48^5.
351060139210 = 8^6 + 50^6 + 53^6 + 56^6 + 81^6
= 18^6 + 43^6 + 49^6 + 64^6 + 80^6
= 27^6 + 31^6 + 56^6 + 62^6 + 80^6
= 35^6 + 36^6 + 38^6 + 73^6 + 76^6.
Links
- R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981.
Extensions
a(5) from Robert Israel, Apr 07 2021
a(6) from Michael S. Branicky, May 08 2021