A175393 a(n) = number of pairs (j,k), j <= k, of divisors of 2n such that j+k divides 2n.

1, 2, 3, 3, 2, 7, 2, 4, 5, 5, 2, 11, 2, 4, 8, 5, 2, 12, 2, 8, 7, 4, 2, 15, 3, 4, 7, 7, 2, 20, 2, 6, 6, 4, 5, 20, 2, 4, 6, 11, 2, 17, 2, 6, 15, 4, 2, 19, 3, 8, 6, 6, 2, 17, 5, 10, 6, 4, 2, 33, 2, 4, 13, 7, 4, 15, 2, 6, 6, 12, 2, 28, 2, 4, 13, 6, 4, 15, 2, 14, 9, 4, 2, 29, 4, 4, 6, 8, 2, 38, 5, 6, 6, 4, 4

Offset

1,2

Comments

The sum of any two divisors of an odd number n never divides n.

From Robert Israel, Oct 09 2017: (Start)

(1,1) and (n,n) are always such pairs, so a(n) >= 2 for n >= 2.

a(n)=2 if and only if n is a prime other than 3.

a(3^k) = 2*k+1.

a(p^k) = k+1 for primes other than 3. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

The divisors of 2*6 = 12 are 1,2,3,4,6,12. The pairs of these divisors that each sum to a divisor of 12 are 1+1 = 2, 1+2 = 3, 1+3 = 4, 2+2 = 4, 2+4 = 6, 3+3 = 6, and 6+6 = 12. There are seven of these pairs, so a(6) = 7.

Maple

f:= proc(n) local D, k, t;
   D:= numtheory:-divisors(2*n);
   t:= 0;
   for k from 1 to nops(D) do
     t:= t + nops(select(j -> 2*n mod (D[j]+D[k])=0, [$1..k]))
   od;
   t
end proc:
map(f, [$1..100]); # Robert Israel, Oct 09 2017

Mathematica

Table[Length[Select[Total/@Union[Sort/@Tuples[Divisors[2 n], 2]], Mod[2 n, #]==0&]], {n, 100}] (* Harvey P. Dale, Apr 09 2023 *)

Crossrefs

Sequence in context: A319442 A299772 A304311 * A184829 A338307 A153290

Adjacent sequences: A175390 A175391 A175392 * A175394 A175395 A175396

Keyword

nonn

Author

Leroy Quet, Apr 29 2010

Extensions

More terms from D. S. McNeil, May 09 2010

Status

approved