A376219 Positions of records in A174414.

1, 10, 11, 33, 88, 132, 341, 505, 1111, 2828, 10504, 31512, 34138, 81103, 152207, 304414, 1378751, 2587519, 2757502, 5175038, 126845092, 486699103, 883779391, 973398206, 1857177597, 1942660159, 3095295995, 7307153656

Offset

1,2

Comments

Positive integers k such that the least m for which the concatenation (m+k)||m is prime is greater than it is for all previous k.

If k and 10^d+1 are not coprime, then A174414(k) can't have d digits.

Therefore, assuming A174414(n) always exists, it is unbounded and this sequence is infinite.

Links

Table of n, a(n) for n=1..28.

Example

a(3) = 11 because A174414(11) = 19 is greater than A174414(1),..., A174414(10).

Maple

tcat:= proc(a, b) a*10^(1+ilog10(b))+b end proc:
f:= proc(n) local k, d;
    for d from 1 do
      if igcd(n, 10^d+1) > 1 then next fi;
      for k from 10^(d-1)+`if`(d=1, 0, 1) to 10^d by 2 do
        if isprime(tcat(n+k, k)) then return k fi
    od od
end proc:
J:= NULL: m:= 0:
for n from 1 to 10^6 do
   v:= f(n);
   if v > m then m:= v; J:= J, n fi
od:
J;

Prog

(Python)
from itertools import count, islice
from math import gcd
from sympy import isprime
def A376219_gen(): # generator of terms
    c = 0
    for n in count(1):
        for l in count(1):
            if gcd(n, (m:=10**l)+1)==1:
                r = m//10
                a = m*(n+r)+r
                for k in range(r, m):
                    if isprime(a):
                        if k>c:
                            yield n
                            c = k
                        break
                    a += m+1
                else:
                    continue
                break
A376219_list = list(islice(A376219_gen(), 30)) # Chai Wah Wu, Sep 18 2024

Crossrefs

Cf. A174414, A376220.

Sequence in context: A041208 A041485 A041206 * A339078 A097990 A280203

Adjacent sequences: A376216 A376217 A376218 * A376220 A376221 A376222

Keyword

nonn,base,more

Author

Robert Israel, Sep 16 2024

Extensions

a(23)-a(28) from Chai Wah Wu, Sep 20 2024

Status

approved