A217049 - OEIS

%I #48 Aug 26 2021 08:54:15

%S 18329,1913,1063,109,932839,85061,29729989,5653759,1958731,20891539,

%T 35008723,28265837,2,3,6238777,276624683,92343187,24205651,49598321,

%U 17722981,46741657,219329923,297614029,106791577,621528749,217893821,113824657,122670287,81263857

%N Least prime p such that all ten base-10 digits have prime counts in (p^prime(n))*(q^prime(n+1)), where q is the next prime after p.

%C Motivation for this sequence stems from the coincidence that (2^41)*(3^43) and (3^43)*(5^47) give prime counts for their digits.

%e (18329^2)*(18341^3) = 2072748335390985614861 has digit counts [2,2,2,3,2,2,2,2,3,2], all primes, and replacing the pair (18329,18341) with a smaller pair fails this criterion. In particular, (3733^2)*(3739^3) = 728420861672094091 has digit counts [3,2,3,0,2,0,2,2,2,2], not all prime.

%t Table[p=2;While[!And@@PrimeQ[DigitCount[(p^Prime@n)*(NextPrime@p^Prime[n+1])]],p=NextPrime@p];p,{n,6}] (* _Giorgos Kalogeropoulos_, Aug 18 2021 *)

%o (Python)

%o from sympy import isprime, nextprime, prime

%o from sympy.ntheory import count_digits

%o def a(n):

%o     pn = prime(n); qn = nextprime(pn)

%o     p, q = 2, 3; c = count_digits(p**pn*q**qn)

%o     while not all(isprime(c[i]) for i in range(10)):

%o         p, q = q, nextprime(q); c = count_digits(p**pn*q**qn)

%o     return p

%o print([a(n) for n in range(1, 7)]) # _Michael S. Branicky_, Aug 20 2021

%Y Cf. A216854.

%K nonn,base

%O 1,1

%A _James G. Merickel_, Sep 25 2012

%E Name clarified by _Tanya Khovanova_, Aug 17 2021

%E a(23)-a(29) from _Michael S. Branicky_, Aug 25 2021

%E Name edited by _Michel Marcus_ and _Michael S. Branicky_, Aug 25 2021