A375938 - OEIS

%I #19 Sep 06 2024 20:16:08

%S 4477,5698,110297,116809,183557,216931,220594,233618,417175,483923,

%T 567358,634106,684167,717541,772079,834350,867724,984533,19940965,

%U 43870123,51846509,52721966,67799281,91728439,103693018,105443932,115657597,131804915,139586755,151551334,171492299,183456878,195421457,207386036

%N Numbers k not divisible by 3 such that no prime is the concatenation of two numbers whose sum is n.

%C Numbers k not divisible by 3 such that A087555(k) = 0.

%C If x has d digits, the concatenation of k - x and x is divisible by GCD(k, 10^d-1). Thus if k has m digits and GCD(k, 10^d-1) > 1 for all d from 2 to m, the only possibility for the concatenation of k - x and x to be prime is when x has a single digit.

%H Michael S. Branicky, <a href="/A375938/b375938.txt">Table of n, a(n) for n = 1..136</a>

%e a(3) = 110297 is a term: 110297 = 11 * 37 * 271 so GCD(110297, 10^d-1) > 1 for d = 2,3,4,5,6. The only possible x for which the concatenation of 110297 - x and x might be prime are the single digits 1, 3, 7 and 9, but none of 1102961, 1102943, 1102907 and 1102889 are prime.

%p tcat:= (a,b) -> a*10^(1+ilog10(b))+b:

%p filter:= proc(n) local d,x;

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

%p   for d from 1 to 1+ilog10(n) do

%p     if igcd(n,10^d-1) > 1 then next fi;

%p     for x from `if`(d=1, 1, 10^(d-1)+1) to 10^d-1 by 2 do

%p       if isprime(tcat(n-x,x)) then return false fi;

%p     od

%p   od;

%p   true

%p end proc:

%p select(filter, [$2 .. 2 * 10^8]);

%o (Python)

%o from gmpy2 import is_prime, mpz, digits

%o def ok(n): return n%3!=0 and not any(is_prime(mpz(digits(n-j)+digits(j))) for j in range(1, n, 2))

%o print([k for k in range(2, 10**6) if ok(k)]) # _Michael S. Branicky_, Sep 03 2024

%Y Cf. A087555.

%K nonn,base

%O 1,1

%A _Robert Israel_, Sep 03 2024