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A333707
Number of ways to write n as the sum of two distinct positive integers that have the same number of divisors.
1
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, 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, 3, 7, 3, 10, 3, 16, 8, 6, 4, 8, 6, 11, 5, 11, 4, 8, 5
OFFSET
1,13
FORMULA
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).
EXAMPLE
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).
MATHEMATICA
Table[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, n - i]], {i, Floor[(n - 1)/2]}], {n, 100}]
CROSSREFS
Cf. A000005, A333626 (not necessarily distinct).
Sequence in context: A117046 A268192 A362312 * A077653 A077889 A305805
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 02 2020
STATUS
approved