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

Author

Michael Somos, Jun 07 2002

Keywords

• nonn
• easy

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

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

Crossrefs

Row sums of A072030. Cf. A000005.

Programs

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 *)