A388051 Integers which can be represented as a sum of two nontrivial powers in two or more ways.

36, 40, 41, 52, 57, 68, 72, 85, 89, 108, 113, 125, 129, 130, 132, 136, 137, 148, 152, 153, 157, 160, 164, 177, 185, 200, 205, 221, 225, 228, 232, 233, 241, 250, 252, 260, 265, 269, 272, 288, 292, 297, 305, 316, 321, 324, 325, 337, 338, 340, 351, 353, 356, 360

Offset

1,1

Comments

The first integer using a power with a composite base is 40.

The first integer where all powers have a composite base is 360.

The first integer represented in three ways is 153.

The first integer to include a non-distinct representation is 72.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Example

40 = 2^2 + 6^2 = 2^3 + 2^5.
360 = 6^2 + 18^2 = 6^3 + 12^2.
153 = 2^5 + 11^2 = 2^7 + 5^2 = 3^2 + 12^2.
468 = 3^3 + 21^2 = 5^3 + 7^3 = 12^2 + 18^2 = 15^2 + 3^5.
1625 = 2^8 + 37^2 = 5^2 + 40^2 = 5^4 + 10^3 = 20^2 + 35^2 = 28^2 + 29^2.
72 = 2^3 + 2^6 = 6^2 + 6^2.

Prog

(PARI)
b(n)=sum(k=2, sqrtint(n\2), if(!ispower(k), sum(e=2, logint(n\2, k), ispower(n-k^e)!=0 )))
select(k->b(k)>=2, [1..500]) \\ Andrew Howroyd, Oct 14 2025

Crossrefs

Cf. A001597, A225103, A298591, A365295, A389486.

Sequence in context: A248372 A349705 A129288 * A389486 A083248 A360765

Adjacent sequences: A388048 A388049 A388050 * A388052 A388053 A388054

Keyword

easy,nonn

Author

Paul Duckett, Oct 14 2025

Status

approved