A360736 Number of prime divisors of A007942(n) = decimal concatenation of sequence (n, n-1, ..., 2, 1, 2, ..., n-1, n) counted with multiplicity.

0, 3, 3, 2, 5, 8, 3, 4, 3, 3, 3, 5, 1, 4, 6, 3, 2, 3, 4, 11, 4, 8, 2, 4, 5, 6, 5, 9, 5, 6, 6, 4, 5, 7, 4, 8, 8, 5, 7, 7, 3, 3, 7, 9, 7, 7, 10, 8, 6, 7, 7, 10, 5, 5

Offset: 1

Author

Bernard Schott, Mar 18 2023

Keywords

• nonn
• base
• more

Comments

For n <= 1530, only a(13) = 1 (answer to Smaradanche problem 19).

First semiprimes appear in A007942 at indices 4, 17, 23 since a(4) = a(17) = a(23) = 2.

Examples

			a(4) = 2 since 4321234 = 2 * 2160617;
a(6) = 8 since 65432123456 = 2^6 * 7 * 146053847;
a(12) = 5 since 12111098765432123456789101112 = 2^3*60800821*24899126702236725259;
a(13) = 1 since 131211109876543212345678910111213 is prime.
		

Links

• M. Fleuren, Factoring of the Smarandache Mirror Sequence.
• F. Smarandache, Only Problems, Not Solutions!, Mirror sequence, problem 19, page 20.

Crossrefs

Cf. A001222, A007942, A110760, A361624.

Programs

Python

from sympy import factorint
def A360736(n): return sum(factorint(int(''.join(map(str,range(n,1,-1)))+''.join(map(str,range(1,n+1))))).values()) # Chai Wah Wu, Mar 21 2023

Formula

a(n) = A001222(A007942(n)).

Extensions

a(36)-a(54) from Amiram Eldar, Mar 19 2023