cp's OEIS Frontend

A305827 Beastly primes with beastly prime indices, i.e., primes p such that both the decimal representation of p and the decimal representation of the index of p in A000040 contain the substring "666".

%I A305827 #23 Jun 04 2025 18:12:42
%S A305827 836663,1266611,5466607,10036661,10856669,13006667,17206667,17256667,
%T A305827 18256669,19666271,22466603,22466617,22466621,22466623,22466629,
%U A305827 26036663,26668007,26716667,30666001,34876661,36668171,39666181,39876667,40666103,41666969,42666667
%N A305827 Beastly primes with beastly prime indices, i.e., primes p such that both the decimal representation of p and the decimal representation of the index of p in A000040 contain the substring "666".
%C A305827 Let p be a prime and let i be the index of p in A000040. Then p is a term of this sequence if both p and i are terms of A051003, or, equivalently, if i is a term of A051003 and p is a term of A131645.
%C A305827 Conjecture 1: The sequence is infinite.
%C A305827 An argument in favor of the conjecture goes as follows: Choose any sufficiently large integer i containing the substring "666". Then it seems very likely, assuming the digits of prime(i) are distributed uniformly, that prime(i) contains the substring "666" at least once.
%C A305827 Based on that argument, one can make the following stronger conjecture:
%C A305827 Conjecture 2: There exists an integer h such that for each i >= h, prime(A051003(i)) is a term of the sequence.
%C A305827 This conjecture may be false if some subsequences of the primes are infinite. For example, assume A004022 is infinite. Then it seems likely that there exists a large term of that sequence whose index in A000040 contains the substring "666".
%e A305827 836663 is prime(66651). Both the prime and its prime index contain the substring "666" in their decimal representation, so 836663 is a term of the sequence.
%t A305827 Select[Prime[Range[26*10^5]],SequenceCount[IntegerDigits[#],{6,6,6}]>0&&SequenceCount[IntegerDigits[PrimePi[#]],{6,6,6}]>0&] (* _Harvey P. Dale_, Jun 04 2025 *)
%o A305827 (PARI) is(n) = my(bst=[6, 6, 6], d=digits(n), e=digits(primepi(n))); for(x=1, #e-2, my(v=[e[x], e[x+1], e[x+2]]); if(v==bst, for(y=1, #d-2, my(w=[d[y], d[y+1], d[y+2]]); if(w==bst, return(1))))); 0
%o A305827 forprime(p=1, , if(is(p), print1(p, ", ")))
%Y A305827 Cf. A000040, A004022, A006450, A051003, A131645.
%K A305827 nonn,base,less
%O A305827 1,1
%A A305827 _Felix Fröhlich_, Jun 10 2018
%E A305827 a(7)-a(26) from _Giovanni Resta_, Jun 11 2018