A190138 Final number of terms obtained with Euler's recurrence formula when computing the sum of divisors of n.

1, 2, 3, 5, 9, 15, 27, 46, 80, 138, 238, 413, 713, 1235, 2136, 3695, 6393, 11057, 19130, 33091, 57246, 99032, 171315, 296365, 512682, 886902, 1534266, 2654154, 4591475, 7942870, 13740526, 23769981, 41120131, 71134474, 123056829, 212878289, 368262059, 637063333

Offset

1,2

Comments

It appears that a(n) is the number of compositions of n whose parts are pentagonal numbers. See Neville link. - Michel Marcus, Jul 28 2017

Links

Table of n, a(n) for n=1..38.

Leonhard Euler, Jordan Bell, A demonstration of a theorem on the order observed in the sums of divisors, arXiv:math/0507201 [math.HO], 2005-2009.

Leonhard Euler, Jordan Bell, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009.

N. Robbins, On compositions whose parts are polygonal numbers, Annales Univ. Sci. Budapest., Sect. Comp. 43 (2014) 239-243. See p. 242.

Example

For n=5, start with row 5 of A195310: [4, 3, 0]. Then replace 4 by row 4: [3, 2], replace 3 by row 3: [2, 1]. The row is now [3, 2, 2, 1, 0].
Repeat process until all terms are 0: [4, 3, 0], [3, 2, 2, 1, 0], [2, 1, 1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0].
The final array has 9 items, hence a(5) = 9.

Mathematica

rows = 30;
gpenta[n_] := If[EvenQ[n], n(3n/2+1)/4, (n+1)(3n+1)/8];
T[n_, k_] := n - gpenta[k];
Do[row[n] = DeleteCases[Table[T[n, k], {k, n}], _?Negative], {n, rows}];
a[n_] := a[n] = row[n] //. j_?Positive :> Sequence @@ row[j] // Length;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, rows}] (* Jean-François Alcover, Sep 22 2018 *)

Prog

(PARI)
A001318(n) = { return((3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8); }
A195310(n) = {if (n == 0, return ([0])); nb = 1; vec = vector(0); nn = n; while (nn >=0, nn = n - A001318(nb); if (nn >=0, vec = concat(vec, nn)); nb++; ); return(vec); }
A190138(m) = { vval = vector(m); for (n=1, m, vec = A195310(n); svec = 0; for (k=1, length(vec), if (vec[k] == 0, svec += 1, svec += vval[vec[k]]); ); vval[n] = svec; ); for (n=1, m, print1(vval[n], ", "); ); }

Crossrefs

Cf. A000203, A001318, A195310.

Sequence in context: A367666 A065954 A067847 * A273905 A307074 A293855

Adjacent sequences: A190135 A190136 A190137 * A190139 A190140 A190141

Keyword

nonn

Author

Michel Marcus, Dec 19 2012

Status

approved