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A358596
a(n) is the least prime p such that the concatenation p|n has exactly n prime factors with multiplicity.
1
3, 2, 83, 2, 67, 41, 947, 4519, 15659081, 2843, 337957, 389, 1616171, 6132829, 422116888343, 24850181, 377519743, 194486417892947, 533348873, 324403, 980825013273164555563, 25691144027, 273933405157, 1238831928746353181, 311195507789, 129917586781, 2159120477658983490299
OFFSET
1,1
COMMENTS
From Jon E. Schoenfield, Feb 26 2023: (Start)
For 2-digit indices n, the following rules can be applied to expedite the search for a(n):
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):
If n is odd, then 2 cannot appear among the prime factors of P(n).
If n == 2 (mod 4), then 2 must appear with multiplicity exactly 1.
If n == 4 (mod 8), then 2 must appear with multiplicity >= 3.
If n == 0 (mod 8), then 2 must appear with multiplicity exactly 2.
If n != 0 (mod 5), then 5 cannot appear among the prime factors of P(n).
If n == 0 (mod 5) but n != 0 (mod 25), then 5 must appear with multiplicity 1.
If n == 0 (mod 25), then 5 must appear with multiplicity >= 2.
Any prime q that divides n but does not divide 10 cannot appear among the prime factors of P(n).
For example, for n = 24, the following constraints apply to the primes that appear among the 24 prime factors of P(24):
since 8 | n, exactly two are 2's;
since 5 !| n, none are 5's;
since 3 | n, none are 3's;
so P(24) >= 2^2 * 7^22. As it turns out, P(24) = 123883192874635318124 = 2^2 * 7^19 * 11 * 13 * 19. (End)
a(924) = floor(2^2 * 13^907 * 17^9 * 19^5 * 31 / 1000) = 8.0881...*10^1026. - Jon E. Schoenfield, Mar 03 2023
LINKS
FORMULA
A001222(A066686(a(n),n)) = n.
EXAMPLE
a(5) = 67 because 67 is prime and 675 = 3^3 * 5^2 with A001222(675) = 3+2 = 5, and no smaller prime works.
MAPLE
icat:= proc(a, b) 10^(1+ilog10(b))*a+b end proc:
f:= proc(n) local p;
p:= 1;
do
p:= nextprime(p);
if numtheory:-bigomega(icat(p, n)) = n then return p fi;
od
end proc:
map(f, [$1..17]); # Robert Israel, Feb 24 2023
CROSSREFS
Sequence in context: A069576 A249678 A300854 * A323745 A109899 A002297
KEYWORD
nonn,base
AUTHOR
Zak Seidov and Robert Israel, Feb 24 2023
EXTENSIONS
a(18) and a(21)-a(27) from Jon E. Schoenfield, Feb 24 2023
STATUS
approved