A131645 -id:A131645 - 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".

836663, 1266611, 5466607, 10036661, 10856669, 13006667, 17206667, 17256667, 18256669, 19666271, 22466603, 22466617, 22466621, 22466623, 22466629, 26036663, 26668007, 26716667, 30666001, 34876661, 36668171, 39666181, 39876667, 40666103, 41666969, 42666667

Offset: 1

Felix Fröhlich, Jun 10 2018

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.
Conjecture 1: The sequence is infinite.
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.
Based on that argument, one can make the following stronger conjecture:
Conjecture 2: There exists an integer h such that for each i >= h, prime(A051003(i)) is a term of the sequence.
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".

			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.

Cf. A000040, A004022, A006450, A051003, A131645.

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 *)

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
forprime(p=1, , if(is(p), print1(p, ", ")))

a(7)-a(26) from Giovanni Resta, Jun 11 2018

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".

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