A375614 Lexicographically earliest infinite sequence of distinct nonnegative pairs of terms that interpenetrate to produce a prime number.

0, 11, 3, 17, 2, 23, 6, 13, 1, 21, 4, 19, 7, 27, 5, 33, 8, 39, 103, 10, 153, 20, 107, 12, 131, 15, 109, 16, 111, 26, 113, 24, 101, 30, 119, 14, 123, 25, 117, 29, 141, 22, 127, 18, 133, 31, 129, 28, 121, 34, 169, 36, 137, 32, 167, 38, 147, 35, 171, 43, 157, 37, 9, 41, 149, 44, 159, 55, 139, 46, 151, 45, 163, 48, 173, 42, 143, 51, 187, 49, 177, 52, 183, 50, 161, 54, 179, 47, 189

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Aug 22 2024

Keywords

• base
• nonn
• look

Comments

The term a(n) must always be exactly one digit longer or shorter than the term a(n+1).

Examples

			Interpenetrate a(1) = 0 and a(2) = 11 to form 101 (a prime number);
interpenetrate a(2) = 11 and a(3) = 3 to form 131 (a prime number);
interpenetrate a(3) = 3 and a(4) = 17 to form 137 (a prime number);
interpenetrate a(4) = 17 and a(5) = 2 to form 127 (a prime number);
interpenetrate a(5) = 2 and a(6) = 23 to form 223 (a prime number);
interpenetrate a(6) = 23 and a(7) = 6 to form 263 (a prime number);
interpenetrate a(7) = 6 and a(8) = 13 to form 163 (a prime number);
interpenetrate a(8) = 13 and a(9) = 1 to form 113 (a prime number);
(...)
interpenetrate a(18) = 39 and a(19) = 103 to form 13093 (a prime number);
(...)
interpenetrate a(167) = 277 and a(168) = 1009 to form 1207079 (a prime number); etc.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Eric Angelini, Combs and Pharaohs, personal blog of the author.

Crossrefs

Cf. A213457, A000040.

Programs

Maple

Q:= proc(a,b) local La, Lb, i;
  La:= convert(a,base,10);
  Lb:= convert(b,base,10);
  add(La[i]*10^(2*i-2),i=1..nops(La)) + add(Lb[i]*10^(2*i-1),i=1..nops(Lb))
end proc:
f:= proc(n) local d,x;
  d:= 1+ilog10(n);
  if n::odd then
    for x from 10^(d-2) to 10^(d-1) - 1 do
      if not(member(x,S)) and isprime(Q(n,x)) then return x fi
    od
  fi;
  for x from 10^d+1 to 10^(d+1) - 1 by 2 do
    if not(member(x,S)) and isprime(Q(x,n)) then return x fi
  od;
FAIL
end proc:
R:= 0,11: S:= {0,11}: v:= 11:
for i from 2 to 100 do
  v:= f(v);
  R:= R,v;
  S:= S union {v};
od:
R; # Robert Israel, Aug 22 2024

Python

from sympy import isprime
from itertools import islice
def ip(s, t): return int("".join(x+v for x, v in zip(s, t))+s[-1])
def agen(): # generator of terms
    seen, an, found = set(), 0, True
    while found:
        yield an
        seen.add(an)
        s = str(an)
        d, found = len(s), False
        if s[-1] in "1379" and d > 1:
            for k in range(10**(d-2), 10**(d-1)):
                if k not in seen and isprime(ip(s, str(k))):
                    an, found = k, True
                    break
        if not found:
            for k in range(10**d, 10**(d+1)):
                if k not in seen and isprime(ip(str(k), s)):
                    an, found = k, True
                    break
print(list(islice(agen(), 90))) # Michael S. Branicky, Aug 22 2024