A328948 - OEIS

A328948

Number of primes that are a concatenation of two positive integers whose product is n.

1, 0, 2, 1, 0, 2, 2, 0, 1, 1, 0, 1, 2, 0, 2, 0, 0, 2, 1, 0, 3, 1, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, 0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 1, 2, 0, 1, 2, 0, 3, 1, 0, 2, 0, 0, 3, 1, 0, 1, 0, 0, 4, 1, 0, 3, 1, 0, 2, 2, 0, 1, 1, 0, 1, 2, 0, 3, 1, 0, 2, 2, 0, 3, 0, 0, 1, 2, 0, 1, 3, 0, 3, 1, 0, 0

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OFFSET

1,3

COMMENTS

Records: 1, 3, 21, 63, 231, 924, 4389, 5187, 51051, 69069, 127281, 245973, 302841, 969969, 1312311, 1716099. - Corrected by Robert Israel, Dec 14 2023

This is not always the same as the number of divisors d of n such that the concatenation of d and n/d is prime, because the same prime could occur for more than one divisor. For example, 1140678 = 14*81477 = 14814*77 with 1481477 prime, and this prime is counted only once in a(1140678) = 7. - Robert Israel, Dec 14 2023

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

a(3n + 2) = 0.

EXAMPLE

1(11), 2(-), 3(13, 31), 4(41), 5(-), 6(23, 61), 7(17, 71), 8(-), 9(19), 10(101), 11(-), 12(43), 13(113, 131), 14(-), 15(53, 151), 16(-).

MAPLE

f:= proc(n)

if n mod 3 = 2 then return 0 fi;

nops(select(isprime, {seq(dcat(t, n/t), t = numtheory:-divisors(n))})

end proc:

map(f, [$1..200]); # Robert Israel, Dec 14 2023