A300836 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 4, 3, 5, 1, 7, 1, 7, 4, 4, 1, 11, 2, 3, 4, 8, 1, 10, 1, 7, 4, 5, 4, 14, 1, 5, 3, 11, 1, 10, 1, 8, 7, 4, 1, 15, 3, 8, 5, 7, 1, 12, 4, 12, 5, 4, 1, 21, 1, 5, 7, 10, 3, 13, 1, 8, 4, 11, 1, 19, 1, 4, 8, 10, 5, 10, 1, 16, 7, 5, 1, 20, 5, 5, 4, 12, 1, 20, 4, 10, 5, 4, 5, 21, 1, 9, 10, 16, 1, 13, 1, 11, 10

Offset: 1

Author

Antti Karttunen, Mar 18 2018

Keywords

• nonn

Examples

			For n=12, its proper divisors are 1, 2, 3, 4 and 6. Zeckendorf-representations (A014417) of these numbers are 1, 10, 100, 101 and 1001. Total number of 1's present is 7, thus a(12) = 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000045, A007895, A014417, A072649, A300834, A300837.

Cf. also A292257, A293435.

Programs

PARI

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A300836(n) = sumdiv(n,d,(dA007895(d));

Formula

a(n) = Sum_{d|n, dA007895(d).

a(n) = A300837(n) - A007895(n).

a(n) = A001222(A300834(n)).

For all n >=1, a(n) >= A293435(n).