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

1,4

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.

Links

Table of n, a(n) for n=1..90.

Formula

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

Example

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.

Prog

(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, k<n && t1(n, k) && t2(n, n/k)) \\ Charles R Greathouse IV, Nov 09 2016

Crossrefs

Cf. A027750, A092506, A277994.

Sequence in context: A127969 A081733 A102587 * A257460 A339471 A159834

Adjacent sequences: A272605 A272606 A272607 * A272609 A272610 A272611

Keyword

nonn

Author

Juri-Stepan Gerasimov, Nov 09 2016

Extensions

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

Status

approved