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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

%I

%S 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,

%T 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,

%U 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

%N 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.

%F 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).

%e 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.

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

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

%K nonn,base

%O 1,5

%A _Wesley Ivan Hurt_, Dec 21 2020

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Last modified May 16 01:55 EDT 2021. Contains 343937 sequences. (Running on oeis4.)