A361624 Number of distinct prime factors in decimal concatenation of integer (n, n-1, ..., 2, 1, 2, ..., n-1, n) = A007942(n).

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

Offset: 1

Author

Bernard Schott, Mar 18 2023

Keywords

• nonn
• base
• more

Comments

a(n) < A360736(n) when n > 10 is a multiple of 4 or of 25, since, for these indices, A007942(n) is divisible by 2^2 or 5^2; but this inequality holds also, for other indices: for n = 6 (see example) and n = 39 where A007942(39) = 29 * 617^2 * 10185403128074353 * ...

Examples

			a(4) = 2 since 4321234 = 2 * 2160617;
a(6) = 3 since 65432123456 = 2^6 * 7 * 146053847.
		

Links

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

Crossrefs

Cf. A001221, A007942, A110760, A360736.

Programs

Python

from sympy import primefactors
def A361624(n): return len(primefactors(int(''.join(map(str,range(n,1,-1)))+''.join(map(str,range(1,n+1)))))) # Chai Wah Wu, Mar 21 2023

Formula

a(n) = A001221(A007942(n)).

Extensions

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

a(42) corrected by Sean A. Irvine, Sep 26 2023

a(55) from Sean A. Irvine, Oct 16 2023