A184219 -id:A184219 - cp's OEIS frontend

A130703 a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

0, 0, 0, 0, 9, 14, 10, 27, 35, 22, 18, 65, 77, 18, 26, 119, 27, 38, 34, 27, 209, 46, 28, 55, 299, 36, 35, 377, 45, 62, 58, 45, 527, 40, 54, 629, 95, 54, 74, 779, 63, 86, 82, 63, 989, 94, 54, 161, 235, 68, 91, 265, 81, 65, 106, 81, 145, 118, 90, 1769, 1829

Offset: 1

Rémi Eismann, Aug 16 2007 - Jan 10 2011

nonn

a(n) is the weight of triangular numbers.

The decomposition of triangular numbers into weight * level + gap is A000217(n) = a(n) * A184219(n) + (n + 1) if a(n) > 0.

			For n = 1 we have A000217(n) = 1, A000217(n+1) = 3; there is no k such that 3 - 1 = 2 = (1 mod k), hence a(1) = 0.
For n = 5 we have A000217(n) = 15, A000217(n+1) = 21; 9 is the smallest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9.
For n = 22 we have A000217(n) = 253, A000217(n+1) = 276; 46 is the smallest k such that 276 - 253 = 23 = (253 mod k), hence a(22) = 46.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368.

A184218 a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377

Offset: 1

Rémi Eismann, Jan 10 2011

From the definition, a(n) = A000217(n) - (n + 1) if A000217(n) - (n + 1) > (n + 1), or 0 otherwise, where A000217 are the triangular numbers.

			For n = 3 we have A000217(3) = 6, A000217(4) = 10; there is no k such that 10 - 6 = 4 = (6 mod k), hence a(3) = 0.
For n = 5 we have A000217(5) = 15, A000217(6) = 21; 9 is the largest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9; a(5) = A000217(5) - (5 + 1) = 15 - 6 = 9.
For n = 24 we have A000217(24) = 300, A000217(25) = 325; 275 is the largest k such that 325 - 300 = 25 = (300 mod k), hence a(24) = 275; a(24) = A000217(24) - (24 + 1) = 275.

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000 [Originally computed by Remi Eismann]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. essentially the same as A000096, A000217, A000027, A130703, A184219, A118534, A117078, A117563, A001223.

[0,0,0,0] cat [(n+1)*(n-2)/2: n in [5..60]]; // Vincenzo Librandi, Jun 22 2016

Join[{0, 0, 0, 0}, LinearRecurrence[{3, -3, 1}, {9, 14, 20}, 100]] (* G. C. Greubel, Jun 22 2016 *)
lim = 10^4; Table[SelectFirst[Reverse@ Range@ lim, Function[k, PolygonalNumber[n + 1] == # + Mod[#, k] &@ PolygonalNumber@ n]], {n, 53}] /. {k_ /; MissingQ@ k -> 0, k_ /; k == lim -> 0} (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)

a(n) = (n+1)*(n-2)/2 = A000096(n-2) for n >= 5 and a(n) = 0 for n <= 4. - M. F. Hasler, Jan 10 2011
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
G.f.: x^5*(5*x^2 - 13*x + 9)/(1 - x)^3. (End)

cp's OEIS Frontend

A130703 a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

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A184218 a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

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