cp's OEIS Frontend

A332101 Least m such that m^n <= Sum_{k

%I A332101 #38 Apr 18 2024 12:50:56
%S A332101 2,3,5,6,8,9,11,12,14,15,16,18,19,21,22,24,25,27,28,29,31,32,34,35,37,
%T A332101 38,40,41,42,44,45,47,48,50,51,52,54,55,57,58,60,61,63,64,65,67,68,70,
%U A332101 71,73,74,76,77,78,80,81,83,84,86,87,89,90,91,93,94,96,97
%N A332101 Least m such that m^n <= Sum_{k<m} k^n.
%C A332101 In a list (1^n, 2^n, 3^n, ...) (rows of table A051128 or A051129), a(n) is the index of the first term less than or equal to the sum of all earlier terms, cf. example.
%C A332101 Obviously a lower bound for any s solution to s^n = Sum_{x in S} x^n, S subset of {1, ..., s-1}, cf. A030052.
%H A332101 Michael De Vlieger, <a href="/A332101/b332101.txt">Table of n, a(n) for n = 0..1000</a>
%H A332101 Brennan Benfield and Oliver Lippard, <a href="https://arxiv.org/abs/2404.08193">Integers that are not the sum of positive powers</a>, arXiv:2404.08193 [math.NT], 2024.
%F A332101 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...)
%F A332101 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
%e A332101 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.
%e A332101 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.
%e A332101 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):
%e A332101   n |k: 1  2   3   4    5    6          Comment
%e A332101   --+---------------------------------------------------------------
%e A332101   1 |  1   2   3                  1 < 2 but 1 + 2 >= 3, so a(1) = 3.
%e A332101   2 |  1   4   9  16   25         1 + 4 + 9 + 16 > 25, and a(2) = 5.
%e A332101   3 |  1   8  27  64  125  216    1 + 8 + 27 + 64 + 125 > 216: a(3) = 6.
%t A332101 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 *)
%o A332101 (PARI) apply( A332101(n,s)=for(m=1,oo, s<m^n||return(m);s+=m^n), [0..66])
%Y A332101 Cf. A051128, A051129.
%Y A332101 Cf. A078607, A332097 (maximum of E(s), cf comments), A030052 (least k such that k^n = sum of distinct n-th powers), A332065 (all k such that k^n is a sum of distinct n-th powers).
%K A332101 nonn
%O A332101 0,1
%A A332101 _M. F. Hasler_, Apr 14 2020