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A007862
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Number of triangular numbers that divide n.
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54
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1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 1, 5, 1, 1, 2, 1, 1, 4, 1, 1, 2, 2, 1, 4, 1, 1, 4, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 5, 1, 1, 3, 1, 1, 4, 1, 1, 2, 2, 1, 4, 1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 5, 1, 1, 2, 1, 1, 6, 2, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 3, 1, 1, 5
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OFFSET
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1,3
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COMMENTS
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Also a(n) is the total number of ways to represent n+1 as a centered polygonal number of the form km(m+1)/2+1 for k>1. - Alexander Adamchuk, Apr 26 2007
a(A130317(n)) = n and a(m) <> n for m < A130317(n). - Reinhard Zumkeller, May 23 2007
Number of oblong numbers that divide 2n. - Ray Chandler, Jun 24 2008
The number of divisors d of 2n such that d+1 is also a divisor of 2n, see first formula. - Michel Marcus, Jun 18 2015
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n forming a finite arithmetic progression with offset 0, i.e. the differences are all equal (with the last part taken to be 0). The Heinz numbers of these partitions are given by A325327. For example, the a(1) = 1 through a(12) = 3 partitions are (A = 10, B = 11, C = 12):
1 2 3 4 5 6 7 8 9 A B C
21 42 63 4321 84
321 642
(End)
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number.
Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
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FORMULA
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a(n) = Sum_{d|2*n,d+1|2*n} 1.
G.f.: Sum_{k>=1} x^A000217(k)/(1-x^A000217(k)). - Jon Perry, Jul 03 2004
a(n) = A129308(2n). - Ray Chandler, Jun 24 2008
a(n) = Sum_{k=1..A000005(n)} A010054(A027750(n,k)). - Reinhard Zumkeller, Jul 05 2014
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MATHEMATICA
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sup=90; TriN=Array[ (#+1)(#+2)/2&, Floor[ N[ Sqrt[ sup*2 ] ] ]-1 ]; Array[ Function[n, 1+Count[ Map[ Mod[ n, # ]&, TriN ], 0 ] ], sup ]
Table[Count[Divisors[k], _?(IntegerQ[Sqrt[8 # + 1]] &)], {k, 105}] (* Jayanta Basu, Aug 12 2013 *)
Table[Length[Select[IntegerPartitions[n], SameQ@@Differences[Append[#, 0]]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)
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PROG
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(Haskell)
a007862 = sum . map a010054 . a027750_row
-- Reinhard Zumkeller, Jul 05 2014
(PARI) a(n) = sumdiv(n, d, ispolygonal(d, 3)); \\ Michel Marcus, Jun 18 2015
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CROSSREFS
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Cf. A046951.
Cf. A010054, A027750, A000005, A239930.
Cf. A000217, A007294, A049988, A325324, A325327, A325407.
Sequence in context: A275699 A305633 A214123 * A285851 A055169 A205131
Adjacent sequences: A007859 A007860 A007861 * A007863 A007864 A007865
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KEYWORD
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nonn
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AUTHOR
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Richard Stanley
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EXTENSIONS
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Extended by Ray Chandler, Jun 24 2008
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STATUS
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approved
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