cp's OEIS Frontend

A083684 Numbers k such that there is no nonnegative integer m such that m < k*prime(k) and the concatenated decimal number fp(k,m) = prime(1).m.prime(2).m. ... .prime(k-1).m.prime(k) is prime.

%I A083684 #24 Apr 10 2020 08:14:01
%S A083684 3,10,16,28,34,40,46,52,70,76,82,88,97,103,121,127,136,163,166,169,
%T A083684 175,187,199,205,211,217,220,235,250,262,268
%N A083684 Numbers k such that there is no nonnegative integer m such that m < k*prime(k) and the concatenated decimal number fp(k,m) = prime(1).m.prime(2).m. ... .prime(k-1).m.prime(k) is prime.
%C A083684 If k == 1 (mod 3) and 3 divides 2 + 3 + 5 + ... + prime(k) then k
%C A083684 is in the sequence. I conjecture that 3 is the only term of the sequence which is not of this form.
%e A083684 For each m, fp(1,m)=2 is prime so 1 is not in the sequence.
%e A083684 fp(2,2) = 2.2.3 = 223 is prime and 2 < 2*prime(2) so 2 isn't in the sequence. Also for each m, 5 divides fp(3,m) = 2.m.3.m.5 so fp(3,m) is composite and we deduce that 3 is in the sequence.
%o A083684 (PARI) is(k) = for(m=0, k*prime(k), if(ispseudoprime(eval(concat(concat([""], vector(2*k-1, i, if(i%2, prime(1+i\2), m)))))), return(0))); 1; \\ _Jinyuan Wang_, Apr 10 2020
%Y A083684 Cf. A082549, A083677.
%K A083684 nonn,base,more
%O A083684 1,1
%A A083684 _Farideh Firoozbakht_, Jun 15 2003
%E A083684 Corrected and edited by _Farideh Firoozbakht_, Nov 04 2013