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%I #46 May 19 2024 03:19:19
%S 2,3,31,443,23053,86677,11827763,27362989,755827199,1306369439
%N a(n) is the smallest prime p such that there exist exactly n distinct primes q where q < p and the representation of p in base q is a palindrome.
%C This is a special case of A372141.
%C It need not be the case that a(n) is a palindrome in base 2, as 23053 is a counterexample.
%C For p > 3, one only needs to check q such that q^2 + 1 <= p else p = cc_q = c*(q+1), not prime for c != 1 and q != 2. A similar argument shows that p cannot have an even number of digits in base q, else it would be divisible by (q+1). - _Michael S. Branicky_, Apr 21 2024
%e a(5) = 86677, as it is palindromic in base 2, 107, 113, 151, and 233, and no smaller number satisfies the property.
%o (Python)
%o from math import isqrt
%o from sympy import sieve
%o from sympy.ntheory import digits
%o from itertools import islice
%o def ispal(v): return v == v[::-1]
%o def f(p): return sum(1 for q in sieve.primerange(1, isqrt(p-1)+1) if ispal(digits(p, q)[1:]))
%o def agen():
%o adict, n = {0:2, 1:3}, 0
%o for p in sieve:
%o v = f(p)
%o if v >= n and v not in adict:
%o adict[v] = p
%o while n in adict:
%o yield adict[n]; del adict[n]; n += 1
%o print(list(islice(agen(), 6))) # _Michael S. Branicky_, Apr 21 2024
%Y Cf. A372141, A002385, A002113.
%Y Cf. A016041, A007500, A077798.
%K nonn,base,more
%O 0,1
%A _Tadayoshi Kamegai_, Apr 21 2024
%E a(6) from _Jon E. Schoenfield_, Apr 21 2024
%E a(7) from _Michael S. Branicky_, Apr 21 2024
%E a(8) from _Michael S. Branicky_, Apr 22 2024
%E a(9) from _Michael S. Branicky_, Apr 24 2024