A110703 -id:A110703 - cp's OEIS frontend

A110701 Friendly run sums: numbers S with two run sums (sum of positive integer runs) that share one common number, i.e., S = a + (a+1) + ... + b = b + (b+1) + ... + c with a < b < c.

9, 21, 30, 42, 65, 70, 99, 105, 117, 133, 135, 154, 175, 180, 225, 231, 275, 285, 341, 342, 345, 364, 385, 414, 440, 450, 455, 481, 495, 540, 546, 567, 630, 645, 675, 693, 744, 750, 765, 825, 833, 936, 945, 990, 1035, 1045, 1140, 1161, 1170, 1176, 1178

Offset: 1

Ron Knott, Aug 04 2005

nonn

The sums are the difference of two triangular numbers A000217. The common numbers themselves are A094550. The sums, n > 0, are n = (b-a+1)(a+b)/2 = (b+c)(c-b+1)/2 where b^2 + a - a^2 = c^2 + c - b^2, is solvable in integers for 0 < a < b < c. Since the runs have something in common, they are "friends". The series of sums *without* the common number is A110702. The numbers common to the two runs are A094550.

			2 + 3 + 4 = 4 + 5 share the common number 4. The sum of both is 9 and this is the smallest such sum with a common "friend" (4), so a(1)=9.

Ron Knott Runsums
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A001227, A094550, A110702, A110703.

A110702 Numbers S with two runsums (sum of positive integer runs) where the 2 runs are separated by a single number gap (b), i.e., S = a + (a+1) + ... + (b-1) = (b+1) + ... + c with a < b-1, b < c.

Original entry on oeis.org

5, 15, 21, 33, 54, 54, 85, 90, 100, 117, 110, 135, 153, 161, 204, 195, 252, 261, 315, 308, 315, 315, 351, 385, 405, 418, 414, 450, 455, 476, 510, 533, 595, 609, 637, 650, 705, 693, 684, 777, 792, 870, 884, 931, 986, 999, 1071, 1105, 1121, 1110, 1125, 1210, 1230

Offset: 1

Ron Knott, Aug 04 2005

nonn

The sums are the difference of two triangular numbers A000217. The sum series where the missing number is included is A110701. The numbers in the gap between the two runs are A094550.

			1+2+3+4+5 and 7+8 have a gap between them of (6). The sum of both is 15 so 15 is in the ordered series (as a(2)).

Ron Knott Runsums
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A001227, A094550, A110701, A110703.

cp's OEIS Frontend

A110701 Friendly run sums: numbers S with two run sums (sum of positive integer runs) that share one common number, i.e., S = a + (a+1) + ... + b = b + (b+1) + ... + c with a < b < c.

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