A333707 - OEIS

%I #10 Apr 20 2020 01:07:59

%S 0,0,0,0,1,0,1,1,1,1,0,1,2,2,1,3,0,3,1,3,2,3,1,4,2,2,1,3,3,5,2,5,2,5,

%T 2,7,2,2,2,6,4,6,4,6,3,5,3,11,5,6,0,6,4,9,3,7,3,4,3,12,7,5,4,10,6,10,

%U 3,7,3,10,3,16,8,6,4,8,6,11,5,11,4,8,5

%N Number of ways to write n as the sum of two distinct positive integers that have the same number of divisors.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=1..floor((n-1)/2)} [d(i) = d(n-i)], where [] is the Iverson bracket and d is the number of divisors of n (A000005).

%e a(16) = 3; There are 3 ways to write 16 as the sum of 2 distinct numbers with the same number of divisors: 16 = 13+3 (13 and 3 both have 2 divisors), 16 = 11+5 (11 and 5 both have 2 divisors), 16 = 10+6 (10 and 6 both have 4 divisors).

%t Table[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, n - i]], {i, Floor[(n - 1)/2]}], {n, 100}]

%Y Cf. A000005, A333626 (not necessarily distinct).

%K nonn,easy

%O 1,13

%A _Wesley Ivan Hurt_, Apr 02 2020