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A190138
Final number of terms obtained with Euler's recurrence formula when computing the sum of divisors of n.
1
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
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
KEYWORD
nonn
AUTHOR
Michel Marcus, Dec 19 2012
STATUS
approved