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Search: id:A110701
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| A110701 |
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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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+0
3
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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 comon, they are "friends". The series of sums *without* the common number is A110702. The numbers common to the two runs are A094550.
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REFERENCES
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T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.
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LINKS
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R. Knott Runsums
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EXAMPLE
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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.
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CROSSREFS
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Cf. A001227, A094550, A110702, A110703.
Sequence in context: A039289 A045252 A139538 this_sequence A133929 A086470 A017629
Adjacent sequences: A110698 A110699 A110700 this_sequence A110702 A110703 A110704
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KEYWORD
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nonn
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AUTHOR
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Ron Knott (enquiry(AT)ronknott.com), Aug 04 2005
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