A072031 Row sums of A072030.

1, 4, 7, 14, 15, 28, 29, 40, 43, 64, 53, 86, 79, 90, 99, 134, 109, 160, 133, 160, 171, 216, 167, 232, 223, 244, 233, 310, 233, 344, 297, 328, 339, 366, 315, 450, 403, 422, 387, 522, 401, 560, 473, 490, 539, 636, 485, 648, 569, 634, 609, 758, 603, 726, 655

Offset

1,2

Comments

From Peter Bala, Dec 09 2014: (Start)

Let A(x) = sum {n >= 1} x^n/(1 - x^n) be the Lambert series generating function of the divisor function tau(n) - see A000005. Then it appears that A(x)^2 is equal to the Lambert series sum {n >= 2} a(n-1)*x^n/(1 - x^n) (checked up to x^1000 using Luschny's formula for a(n)).

This conjecture is equivalent to the following identity for the divisor function: for n >= 2 there holds sum {k = 1..n} tau(k)*tau(n-k) = sum {d divides n} ( sum {k = 1..d} tau(k*(d - k)) ), where we take tau(0) = 0. (End)

Links

Table of n, a(n) for n=1..55.

James C. Alexander, The Entropy of the Fibonacci Code (1989).

Maple

A072031 := n -> add(numtheory[tau](j*(n+1-j)), j=1..n);
seq(A072031(i), i=1..55); # Peter Luschny, Sep 13 2012

Mathematica

T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0, n == k, 1, n < k, T[k, n], True, 1 + T[k, n - k]];
a[n_] := Sum[T[n - k + 1, k],  {k, 1, n}];
Array[a, 55] (* Jean-François Alcover, Jun 03 2019 *)

Crossrefs

Row sums of A072030. A000005.

Sequence in context: A310850 A310851 A310852 * A252788 A310853 A310854

Adjacent sequences: A072028 A072029 A072030 * A072032 A072033 A072034

Keyword

nonn,easy

Author

Michael Somos, Jun 07 2002

Status

approved