A375938 Numbers k not divisible by 3 such that no prime is the concatenation of two numbers whose sum is n.

4477, 5698, 110297, 116809, 183557, 216931, 220594, 233618, 417175, 483923, 567358, 634106, 684167, 717541, 772079, 834350, 867724, 984533, 19940965, 43870123, 51846509, 52721966, 67799281, 91728439, 103693018, 105443932, 115657597, 131804915, 139586755, 151551334, 171492299, 183456878, 195421457, 207386036

Offset

1,1

Comments

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

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.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..136

Example

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.

Maple

tcat:= (a, b) -> a*10^(1+ilog10(b))+b:
filter:= proc(n) local d, x;
  if n mod 3 = 0 then return false fi;
  for d from 1 to 1+ilog10(n) do
    if igcd(n, 10^d-1) > 1 then next fi;
    for x from `if`(d=1, 1, 10^(d-1)+1) to 10^d-1 by 2 do
      if isprime(tcat(n-x, x)) then return false fi;
    od
  od;
  true
end proc:
select(filter, [$2 .. 2 * 10^8]);

Prog

(Python)
from gmpy2 import is_prime, mpz, digits
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))
print([k for k in range(2, 10**6) if ok(k)]) # Michael S. Branicky, Sep 03 2024

Crossrefs

Cf. A087555.

Sequence in context: A221007 A203938 A345181 * A235890 A237775 A179161

Adjacent sequences: A375935 A375936 A375937 * A375939 A375940 A375941

Keyword

nonn,base

Author

Robert Israel, Sep 03 2024

Status

approved