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A376219 Positions of records in A174414.

%I A376219 #33 Sep 21 2024 02:23:36
%S A376219 1,10,11,33,88,132,341,505,1111,2828,10504,31512,34138,81103,152207,
%T A376219 304414,1378751,2587519,2757502,5175038,126845092,486699103,883779391,
%U A376219 973398206,1857177597,1942660159,3095295995,7307153656
%N A376219 Positions of records in A174414.
%C A376219 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.
%C A376219 If k and 10^d+1 are not coprime, then A174414(k) can't have d digits.
%C A376219 Therefore, assuming A174414(n) always exists, it is unbounded and this sequence is infinite.
%e A376219 a(3) = 11 because A174414(11) = 19 is greater than A174414(1),..., A174414(10).
%p A376219 tcat:= proc(a,b) a*10^(1+ilog10(b))+b end proc:
%p A376219 f:= proc(n) local k,d;
%p A376219     for d from 1 do
%p A376219       if igcd(n, 10^d+1) > 1 then next fi;
%p A376219       for k from 10^(d-1)+`if`(d=1,0,1) to 10^d by 2 do
%p A376219         if isprime(tcat(n+k,k)) then return k fi
%p A376219     od od
%p A376219 end proc:
%p A376219 J:= NULL: m:= 0:
%p A376219 for n from 1 to 10^6 do
%p A376219    v:= f(n);
%p A376219    if v > m then m:= v; J:= J,n fi
%p A376219 od:
%p A376219 J;
%o A376219 (Python)
%o A376219 from itertools import count, islice
%o A376219 from math import gcd
%o A376219 from sympy import isprime
%o A376219 def A376219_gen(): # generator of terms
%o A376219     c = 0
%o A376219     for n in count(1):
%o A376219         for l in count(1):
%o A376219             if gcd(n,(m:=10**l)+1)==1:
%o A376219                 r = m//10
%o A376219                 a = m*(n+r)+r
%o A376219                 for k in range(r,m):
%o A376219                     if isprime(a):
%o A376219                         if k>c:
%o A376219                             yield n
%o A376219                             c = k
%o A376219                         break
%o A376219                     a += m+1
%o A376219                 else:
%o A376219                     continue
%o A376219                 break
%o A376219 A376219_list = list(islice(A376219_gen(), 30)) # _Chai Wah Wu_, Sep 18 2024
%Y A376219 Cf. A174414, A376220.
%K A376219 nonn,base,more
%O A376219 1,2
%A A376219 _Robert Israel_, Sep 16 2024
%E A376219 a(23)-a(28) from _Chai Wah Wu_, Sep 20 2024