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 A110703 Numbers S with two neighboring run sums (sum of positive integer runs) S = a+(a+1)+..+b=(b+1)+(b+2)...+c, 0
 3, 15, 27, 30, 42, 75, 90, 105, 135, 147, 165, 243, 252, 270, 273, 315, 363, 375, 378, 420, 462, 495, 507, 612, 660, 675, 693, 735, 750, 780, 810, 855, 858, 867, 945, 1050, 1083, 1155, 1170, 1215, 1287, 1323, 1365, 1470, 1485, 1518, 1587, 1785, 1815, 1875, 1950 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS In other words, numbers n such that a list of consecutive numbers can be split into two parts in which their sums both equal n. - A. D. Skovgaard, May 22 2017 If the two runs overlap in one number, the runs are Friends and their sums are A110701. The sums are the difference of two triangular numbers A000217. The subsequence where there is more than one possible splitting begins 105, 945, 1365, 2457, 2625, 3990, 5145, 8505, ... - Jean-François Alcover, May 22 2017 a(n) seems to always be divisible by 3.- A. D. Skovgaard, May 22 2017. This is true. Sequence lists values of n = t(t+1)/2 - k(k+1)/2 = m(m+1)/2 - t(t+1)/2 with k < t < m. Since any triangular number must be of the form 3w or 3w+1, then there are two possibilities for n = 3w - k(k+1)/2 = m(m+1)/2 - 3w or n = 3w + 1 - k(k+1)/2 = m(m+1)/2 - 3w - 1. For first case, if k(k+1)/2 = 3u+1, there is no solution for m. Similarly for second case, if k(k+1)/2 = 3u, there is no solution for m. So always n must be divisible by 3. - Altug Alkan, May 22 2017 LINKS Ron Knott Runsums T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6. EXAMPLE 3 = 1+2 = 3, so 3 is a term. 15 = 4+5+6 = 7+8 so 15 is a term. a(6) = 75 because 75 = 3+4+5+6+7+8+9+10+11+12 = 13+14+15+16+17. MATHEMATICA Select[Range[1000], False =!= Reduce[# == Sum[k, {k, x, y}] == Sum[k, {k, y + 1, z}] && z >= y >= x > 0, {x, y, z}, Integers] &] (* Giovanni Resta, May 22 2017 *) CROSSREFS Cf. A001227, A094550, A110701, A110702. Sequence in context: A328387 A006872 A027179 * A121250 A105549 A080065 Adjacent sequences:  A110700 A110701 A110702 * A110704 A110705 A110706 KEYWORD nonn AUTHOR Ron Knott, Aug 04 2005 EXTENSIONS Initial 3 added by A. D. Skovgaard, May 22 2017 STATUS approved

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Last modified July 6 12:26 EDT 2022. Contains 355110 sequences. (Running on oeis4.)