A339632 Number of partitions of 2n into two positive integer parts (s,t) such that s and t have the same number of decimal digits and s*t is semiprime.

0, 1, 1, 1, 3, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 3, 2, 2, 4, 2, 3, 5, 2, 3, 4, 1, 4, 5, 3, 3, 5, 3, 4, 7, 3, 3, 8, 3, 4, 6, 3, 5, 7, 3, 4, 6, 4, 4, 6, 3, 3, 7, 2, 2, 6, 2, 3, 4, 3, 2, 3, 3, 3, 3, 2, 1, 4, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 0, 1

Offset

1,5

Links

Table of n, a(n) for n=1..93.

Formula

a(n) = Sum_{k=1..n} [Omega(k*(2*n-k)) = 2] * [floor(log_10(k)) = floor(log_10(2*n-k))], where [ ] is the Iverson bracket and Omega is the number of prime factors of n with multiplicity (A001222).

Example

a(18) = 0; 18 has 9 partitions into two positive integer parts, (17,1), (16,2), (15,3), (14,4), (13,5), (12,6), (11,7), (10,8), (9,9). There are no partitions whose parts have the same number of decimal digits and whose product is semiprime.

Mathematica

Table[Sum[KroneckerDelta[PrimeOmega[i (2 n - i)], 2] KroneckerDelta[ Floor[Log10[i]], Floor[Log10[2 n - i]]], {i, n}], {n, 100}]

Crossrefs

Cf. A001222 (Omega), A001358, A055642, A078972.

Sequence in context: A074063 A338211 A115717 * A350879 A115718 A361510

Adjacent sequences: A339629 A339630 A339631 * A339633 A339634 A339635

Keyword

nonn,base

Author

Wesley Ivan Hurt, Dec 21 2020

Status

approved