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

1,2

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.

Links

Table of n, a(n) for n=1..54.

M. Fleuren, Factoring of the Smarandache Mirror Sequence.

F. Smarandache, Only Problems, Not Solutions!, Mirror sequence, problem 19, page 20.

Formula

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

Example

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.

Prog

(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

Crossrefs

Cf. A001222, A007942, A110760, A361624.

Sequence in context: A290599 A070163 A083343 * A292527 A359135 A359484

Adjacent sequences: A360733 A360734 A360735 * A360737 A360738 A360739

Keyword

nonn,base,more

Author

Bernard Schott, Mar 18 2023

Extensions

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

Status

approved