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
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
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Jun 07 2002
STATUS
approved