OFFSET

0,1

COMMENTS

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1000

Brennan Benfield and Oliver Lippard, Integers that are not the sum of positive powers, arXiv:2404.08193 [math.NT], 2024.

FORMULA

a(n) = round(n / log(2)) + 2. (Conjectured; verified up to 10^4, in particular for 3525/log(2) = 5085.500019... and 7844/log(2) ~ 11316.49990...)

a(n) = A078607(n) + 2 for almost all n > 1. (n = 777451915729368 might be an exception to this equality or the above one.) - M. F. Hasler, May 08 2020

EXAMPLE

For n = 0, m^0 > Sum_{0 < k < m} k^0 = 0 for m = 0, 1 (empty sums), but 2^0 = Sum_{0 < k < 2} k^0 = 1, so a(0) = 2.

For n = 1, 1^1 > Sum_{0 < k < 1} k^1 = 0 (empty sum) and 2^1 > Sum_{0 < k < 2} k^1 = 1, but 3^1 <= Sum_{0 < k < 3} k^1 = 1 + 2, so a(1) = 3.

To find a(n) one can add up terms in row n of the table k^n until the sum equals or exceeds the next term, whose column number k is then a(n):

n |k: 1 2 3 4 5 6 Comment

--+---------------------------------------------------------------

1 | 1 2 3 1 < 2 but 1 + 2 >= 3, so a(1) = 3.

2 | 1 4 9 16 25 1 + 4 + 9 + 16 > 25, and a(2) = 5.

3 | 1 8 27 64 125 216 1 + 8 + 27 + 64 + 125 > 216: a(3) = 6.

MATHEMATICA

Table[Block[{m = 1, s = 0}, While[m^n > s, s = s + m^n; m++]; m], {n, 0, 66}] (* Michael De Vlieger, Apr 30 2020 *)

PROG

(PARI) apply( A332101(n, s)=for(m=1, oo, s<m^n||return(m); s+=m^n), [0..66])

CROSSREFS

KEYWORD

nonn

AUTHOR

M. F. Hasler, Apr 14 2020

STATUS

approved