cp's OEIS Frontend

A093001 Least k such that Sum_{r=n+1..k} r is greater than or equal to the sum of the first n positive integers (i.e., the n-th triangular number, A000217(n)). Or, least k such that (sum of first n positive integers) <= (sum of numbers from n+1 up to k).

%I A093001 #15 Sep 08 2022 08:45:13
%S A093001 2,3,5,6,8,9,11,12,13,15,16,18,19,20,22,23,25,26,28,29,30,32,33,35,36,
%T A093001 37,39,40,42,43,45,46,47,49,50,52,53,54,56,57,59,60,62,63,64,66,67,69,
%U A093001 70,71,73,74,76,77,78,80,81,83,84,86,87,88,90,91,93,94,95,97,98,100,101
%N A093001 Least k such that Sum_{r=n+1..k} r is greater than or equal to the sum of the first n positive integers (i.e., the n-th triangular number, A000217(n)). Or, least k such that (sum of first n positive integers) <= (sum of numbers from n+1 up to k).
%F A093001 a(n) = ceiling((-1 + sqrt(8n^2 + 8n + 1))/2). - _Joshua Zucker_, May 08 2006
%e A093001 a(6)= 9, 7+8+9 > 1+2+3+4+5+6, but 7+8 < 1+2+3+4+5+6.
%t A093001 Table[Ceiling[(-1 + Sqrt[8 n^2 + 8 n + 1])/2], {n, 0, 80}] (* _Vincenzo Librandi_, Jun 23 2015 *)
%o A093001 (PARI) a(n) = {my(k=1); while (sum(j=n+1, k, j) < n*(n+1)/2, k++); k;} \\ _Michel Marcus_, Jul 23 2015
%o A093001 (Magma) [Ceiling((-1 + Sqrt(8*n^2 + 8*n + 1))/2): n in [1..80]]; // _Vincenzo Librandi_, Jul 23 2015
%Y A093001 Cf. A093000.
%K A093001 easy,nonn
%O A093001 1,1
%A A093001 _Amarnath Murthy_, Mar 29 2004
%E A093001 More terms from _Joshua Zucker_, May 08 2006