cp's OEIS Frontend

A076114 a(n) = start of the smallest string of n consecutive positive integers with a square sum, or 0 if no such number exists.

%I A076114 #19 Aug 19 2025 08:53:02
%S A076114 1,4,2,0,3,11,4,1,5,18,6,0,7,25,8,0,9,4,10,0,11,39,12,2,4,46,14,0,15,
%T A076114 53,16,9,17,60,18,0,19,67,20,3,21,74,22,0,23,81,24,0,1,16,26,0,27,11,
%U A076114 28,4,29,102,30,0,31,109,32,0,33,116,34,0,35,123,36,5,37,130,11,0,39
%N A076114 a(n) = start of the smallest string of n consecutive positive integers with a square sum, or 0 if no such number exists.
%C A076114 Equivalently, smallest k such that n(n+2k-1)/2 is a square, 0 if there is no such square. - _Ralf Stephan_, Mar 23 2003
%C A076114 a(n)=0 if and only if n has the form 4^e*m with e > 0 and m odd. - _Dean Hickerson_, Mar 25 2003
%C A076114 a(k) = 1 if k is the index of a square triangular number, i.e., k(k+1)/2 is a square, or if k belongs to A001108. If n is odd then a(n) <= (n+1)/2.
%H A076114 Clement Martin, <a href="/A076114/b076114.txt">Table of n, a(n) for n = 1..10000</a>
%e A076114 a(2) = 4, 4+5 = 9 = 3^2. a(8) = 1, 1+2+3+4+5+6+7+8 = 36 = 6^2.
%o A076114 (PARI) a(n,limit=oo) = { my(o=n*(n+1)/2, k=0); while(k<limit && !issquare(o+n*k), k++); if(k<limit, k+1, oo) }
%Y A076114 Cf. A001108, A001109, A076115.
%K A076114 nonn,changed
%O A076114 1,2
%A A076114 _Amarnath Murthy_, Oct 09 2002
%E A076114 More terms from _Ralf Stephan_, Mar 23 2003