A272608 Number of positive integers k such that both n/(k + 2^x) and n/(n/k - 2^y) are integers for some nonnegative x, y.

0, 1, 1, 2, 1, 2, 0, 3, 1, 2, 0, 4, 0, 0, 1, 4, 1, 2, 0, 3, 1, 0, 0, 6, 0, 0, 0, 0, 0, 3, 0, 5, 2, 2, 1, 4, 0, 0, 0, 4, 0, 2, 0, 0, 1, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 5, 0, 0, 1, 6, 2, 4, 0, 2, 0, 1, 0, 6, 0, 0, 0, 0, 1, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2

Offset: 1

Author

Juri-Stepan Gerasimov, Nov 09 2016

Keywords

• nonn

Comments

Where k, k + 2^x, n/k, n/k - 2^y, n/(k + 2^x) and n/(n/k - 2^y) are divisors of n.

Examples

			a(9) = 1 because both 9/(1 + 2^1) = 3 and 9/(9/1 - 2^4) = 1 are integers.
a(68) = 3 because (1) 68/(1 + 2^0) = 34 and 68/(68 - 2^6) = 17, (2) 68/(2 + 2^1) = 17 and 68/(34 - 2^5) = 34, and (3) 68/(4 + 2^6) = 1 and 68/(17 - 2^4) = 68 are all integers.
		

Crossrefs

Cf. A027750, A092506, A277994.

Programs

PARI

t1(n,k)=for(x=0,logint(n,2), if(n%(k+2^x)==0, return(1))); 0
t2(n,d)=for(y=0,logint(d-1,2), if(n%(d-2^y)==0, return(1))); 0
a(n)=sumdiv(n,k, kCharles R Greathouse IV, Nov 09 2016

Formula

a(2^n) = n, a(A092506(n)) = 1.

Extensions

a(68), a(70), a(90) corrected by Charles R Greathouse IV, Nov 09 2016