A081175 Numbers of the form Sum_{i=1..k} i^j, j >= 1, k >= 1.

1, 3, 5, 6, 9, 10, 14, 15, 17, 21, 28, 30, 33, 36, 45, 55, 65, 66, 78, 91, 98, 100, 105, 120, 129, 136, 140, 153, 171, 190, 204, 210, 225, 231, 253, 257, 276, 285, 300, 325, 351, 354, 378, 385, 406, 435, 441, 465, 496, 506, 513, 528, 561, 595, 630, 650, 666, 703

Offset: 1

Author

N. J. A. Sloane, Apr 18 2003

Keywords

• nonn
• easy

Comments

Union of sums of k-th powers, for k >= 1.

Examples

			30 is in the set because 30 = 1^2 + 2^2 + 3^2 + 4^2 (j=2, k=4).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Michael Penn, an excruciatingly deep dive into the power sum., YouTube video, 2022.

Crossrefs

Cf. A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542, A007487, A023002.

For primes in this sequence see A164307.

Programs

Maple

N:= 1000: # to get all terms <= N
A:=select(`<=`,{1, seq(seq(sum(i^k,i=1..m), m=2..floor((N*(k+1))^(1/(k+1)))),k = 1 ..ilog2(N-1))},N):
sort(convert(A,list)); # Robert Israel, Jan 26 2015

Mathematica

Take[ Union[ Flatten[ Table[ Sum[ i^j, {i, 1, n}], {j, 1, 9}, {n, 1, 40}]]], 60]

Extensions

Corrected and extended by Robert G. Wilson v, May 08 2003