A260152 - OEIS

%I #28 Aug 29 2018 15:18:51

%S 1,3,3,6,3,9,3,10,6,10,3,16,3,10,9,15,3,19,3,20,10,10,3,26,6,10,10,20,

%T 3,28,3,21,10,10,10,33,3,10,10,32,3,32,3,21,18,10,3,40,6,21,10,21,3,

%U 33,10,32,10,10,3,50,3,10,20,28,10,33,3,21,10,34

%N a(n) is the number of distinct sums s + t where s, t are divisors of n.

%C s and t need not be distinct.

%C a(n) = 3 if and only if n is prime.

%C If p is prime, a(p^k) = A000217(k+1).

%C If p is in A005382, a(p*(2*p-1)) = 9.  For all other members of A006881, a(n) = 10.

%C a(n) <= A000217(A000005(n)).

%H Robert Israel, <a href="/A260152/b260152.txt">Table of n, a(n) for n = 1..10000</a>

%H MathOverflow, <a href="http://mathoverflow.net/questions/223036">A different kind of divisor sums</a>

%e For n = 2 the divisors are 1 and 2, and the a(2) = 3 distinct sums are 1+1=2, 1+2=3, 2+2=4.

%p T:= proc(n) local D,x,y;

%p   D:= numtheory:-divisors(n);

%p   nops({seq(seq(x+y,x=D),y=D)})

%p end proc:

%p seq(T(n),n=1..100);

%o (PARI) a(n) = my(v=[], d = divisors(n)); for (i=1, #d, for (j=i, #d, v = concat(v, d[i]+d[j]))); #Set(v); \\ _Michel Marcus_, Aug 29 2018

%Y Cf. A000005, A000217, A005382, A006881, A007425.

%Y Cf. A048691 (with distinct products s*t rather than sums).

%K nonn

%O 1,2

%A _Robert Israel_, Nov 09 2015