A328948 - OEIS

%I #76 Dec 16 2023 09:00:19

%S 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,

%T 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,

%U 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

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

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

%C 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

%H Robert Israel, <a href="/A328948/b328948.txt">Table of n, a(n) for n = 1..10000</a>

%F a(3n + 2) = 0.

%e 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(-).

%p f:= proc(n)

%p    if n mod 3 = 2 then return 0 fi;

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

%p end proc:

%p map(f, [$1..200]); # _Robert Israel_, Dec 14 2023

%o (PARI) a(n) = sumdiv(n, d, isprime(eval(concat(Str(d), Str(n/d))))); \\ _Michel Marcus_, Nov 05 2019

%o (Magma) [#[a: d in Divisors(n)| IsPrime(a) where a is Seqint(Intseq(d) cat Intseq(n div d))]:n in [1..100]]; // _Marius A. Burtea_, Nov 05 2019

%Y Cf. A000040, A016789, A161904, A328903.

%K nonn,base,easy

%O 1,3

%A _Juri-Stepan Gerasimov_, Nov 01 2019