A361624 - OEIS

%I #30 Oct 16 2023 22:28:30

%S 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,

%T 4,5,8,5,6,6,3,3,7,7,7,7,10,7,6,6,7,4,5,5,7

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

%C 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 * ...

%H M. Fleuren, <a href="http://www.gallup.unm.edu/~smarandache/SmMirror.txt">Factoring of the Smarandache Mirror Sequence</a>.

%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, Mirror sequence, problem 19, page 20.

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

%e a(4) = 2 since 4321234 = 2 * 2160617;

%e a(6) = 3 since 65432123456 = 2^6 * 7 * 146053847.

%o (Python)

%o from sympy import primefactors

%o 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

%Y Cf. A001221, A007942, A110760, A360736.

%K nonn,base,more

%O 1,2

%A _Bernard Schott_, Mar 18 2023

%E a(36)-a(54) from _Amiram Eldar_, Mar 19 2023

%E a(42) corrected by _Sean A. Irvine_, Sep 26 2023

%E a(55) from _Sean A. Irvine_, Oct 16 2023