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%I #31 Sep 08 2022 08:45:13
%S 4,6,9,11,14,15,16,17,19,21,22,23,24,25,26,29,30,31,32,34,35,36,38,39,
%T 40,41,43,44,46,48,49,50,51,52,53,54,56,57,59,61,64,66,68,69,70,71,72,
%U 74,76,77,79,81,82,83,84,86,87,89,91,93,94,95,96,97,98,99,100,101,104
%N Numbers n such that there are integers a < b with a+(a+1)+...+(n-1) = (n+1)+(n+2)+...+b.
%C Liljestrom shows that n is in this sequence if and only if 4n^2+1 is composite.
%C Complement of A001912.
%C From _Hermann Stamm-Wilbrandt_, Sep 16 2014: (Start)
%C For n > 1, A047209 is a subset of this sequence [ 4*n^2+1 is divisible by 5 if n is (1 or 4) mod 5].
%C A092464 is a subset of this sequence [4*n^2+1 is divisible by 13 if n is (4 or 9) mod 13].
%C The above are for divisibility by 5, 13; notation (1,4,5), (4,9,13). Divisibility by p for a and p-a; notation (a,p-a,p). These are the next tuples: (2,15,17), (6,23,29), (3,34,37), (16,25,41), ... . The corresponding sequences are a subset of this sequence [ 2,15,19,32,36,49,... for (2,15,17) ]. These sequences have no entries in the OEIS yet. For any prime of the form 4*k+1 there is exactly one of these tuples/sequences [solution to 4*a^2+1=0 (mod p)].
%C For n>1, A000290 (squares) is a subset of this sequence (4,9,16,25,...) [ 4*(m^2)^2+1 is divisible by m^2+(m+1)^2, tuple (m^2, (m+1)^2, m^2+(m+1)^2) ].
%C (End)
%H Vincenzo Librandi, <a href="/A094550/b094550.txt">Table of n, a(n) for n = 1..1000</a>
%H R. J. Liljestrom and Richard Zucker, <a href="http://www.powershow.com/view1/82b20-ZDc1Z/NUMERICAL_FULCRUMS_powerpoint_ppt_presentation">Numerical Fulcrums (PowerPoint Format)</a>
%e 6 is in this sequence because 1+2+3+4+5 = 7+8.
%t lst={}; Do[i1=n-1; i2=n+1; s1=i1; s2=i2; While[i1>1 && s1!=s2, If[s1<s2, i1--; s1=s1+i1, i2++; s2=s2+i2]]; If[s1==s2, AppendTo[lst, n]], {n, 2, 140}];
%t lst (* end of program *)
%t Select[Range[1000],!PrimeQ[4#^2+1]&] (* _T. D. Noe_, Nov 12 2010 *)
%o (Magma) [n: n in [1..100] |not IsPrime(4*n^2 + 1)]; // _Vincenzo Librandi_, Sep 27 2012 *)
%Y Cf. A094551, A094552, A094553.
%Y Cf. A001912.
%K nonn,easy
%O 1,1
%A _T. D. Noe_, May 10 2004
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