A358596 - OEIS

%I #24 Mar 03 2023 22:35:02

%S 3,2,83,2,67,41,947,4519,15659081,2843,337957,389,1616171,6132829,

%T 422116888343,24850181,377519743,194486417892947,533348873,324403,

%U 980825013273164555563,25691144027,273933405157,1238831928746353181,311195507789,129917586781,2159120477658983490299

%N a(n) is the least prime p such that the concatenation p|n has exactly n prime factors with multiplicity.

%C From _Jon E. Schoenfield_, Feb 26 2023: (Start)

%C For 2-digit indices n, the following rules can be applied to expedite the search for a(n):

%C Let P(n) be the concatenation of a(n) and n. Then P(n) is the product of n primes (counted with multiplicity), P(n) mod 100 = n, a(n) = floor(P(n)/100) is prime, and it can be shown that the following constraints apply to the prime factors of P(n):

%C If n is odd, then 2 cannot appear among the prime factors of P(n).

%C If n == 2 (mod 4), then 2 must appear with multiplicity exactly 1.

%C If n == 4 (mod 8), then 2 must appear with multiplicity >= 3.

%C If n == 0 (mod 8), then 2 must appear with multiplicity exactly 2.

%C If n != 0 (mod 5), then 5 cannot appear among the prime factors of P(n).

%C If n == 0 (mod 5) but n != 0 (mod 25), then 5 must appear with multiplicity 1.

%C If n == 0 (mod 25), then 5 must appear with multiplicity >= 2.

%C Any prime q that divides n but does not divide 10 cannot appear among the prime factors of P(n).

%C For example, for n = 24, the following constraints apply to the primes that appear among the 24 prime factors of P(24):

%C   since 8 | n, exactly two are 2's;

%C   since 5 !| n, none are 5's;

%C   since 3 | n, none are 3's;

%C so P(24) >= 2^2 * 7^22. As it turns out, P(24) = 123883192874635318124 = 2^2 * 7^19 * 11 * 13 * 19. (End)

%C a(924) = floor(2^2 * 13^907 * 17^9 * 19^5 * 31 / 1000) = 8.0881...*10^1026. - _Jon E. Schoenfield_, Mar 03 2023

%H Jon E. Schoenfield, <a href="/A358596/b358596.txt">Table of n, a(n) for n = 1..923</a>

%F A001222(A066686(a(n),n)) = n.

%e a(5) = 67 because 67 is prime and 675 = 3^3 * 5^2 with A001222(675) = 3+2 = 5, and no smaller prime works.

%p icat:= proc(a,b) 10^(1+ilog10(b))*a+b end proc:

%p f:= proc(n) local p;

%p p:= 1;

%p do

%p   p:= nextprime(p);

%p   if numtheory:-bigomega(icat(p,n)) = n then return p fi;

%p od

%p end proc:

%p map(f, [$1..17]); # _Robert Israel_, Feb 24 2023

%Y Cf. A001222, A066686.

%K nonn,base

%O 1,1

%A _Zak Seidov_ and _Robert Israel_, Feb 24 2023

%E a(18) and a(21)-a(27) from _Jon E. Schoenfield_, Feb 24 2023