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

Author

Ron Knott, Aug 04 2005

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A001227, A094550, A110702, A110703.