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 A359840 Numbers k that are the representation of primes in base 4 and in base 5. 0
 2, 3, 23, 131, 133, 221, 1211, 1231, 2023, 2111, 2113, 2311, 3013, 3211, 3233, 3323, 10031, 10033, 10121, 12011, 12121, 13223, 13331, 20131, 20203, 22111, 23233, 31313, 32033, 32303, 33133, 33331, 100123, 100211, 100231, 101003, 101333, 103333, 110021, 111211 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For a(1) = 2, 2_4 = 2_5 = 2_10 and for a(2) = 3, 3_4 = 3_5 = 3_10; otherwise, these two primes are distinct for n >= 3 (example). The corresponding sequences of primes are A235474 (for base 4) and A235615 (for base 5). LINKS Table of n, a(n) for n=1..40. FORMULA a(n) = A007090(A235474(n)); a(n) = A007091(A235615(n)). EXAMPLE a(3) = 23 because 23_4 = 11_10 = A235474(3) and 23_5 = 13_10 = A235615(3) are primes. a(9) = 2023 because 2023_4 = 139_10 = A235474(9) and 2023_5 = 263_10 = A235615(9) are primes. MATHEMATICA q[n_, b_] := Max[d = IntegerDigits[n]] < b && PrimeQ[FromDigits[d, b]]; Select[Range[200000], q[#, 4] && q[#, 5] &] (* Amiram Eldar, Jan 15 2023 *) PROG (Python) from sympy import isprime def ok(n): return max(s:=str(n)) < '4' and isprime(int(s, 4)) and isprime(int(s, 5)) print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 15 2023 (Python) from sympy import isprime from itertools import count, islice, product def agen(): yield from (int(s) for d in count(1) for f in "123" for r in product("0123", repeat=d-1) if isprime(int(s:=f+"".join(r), 4)) and isprime(int(s, 5))) print(list(islice(agen(), 40))) # Michael S. Branicky, Jan 15 2023 CROSSREFS Intersection of A004678 and A004679. Cf. A007090, A007091, A235474, A235615, A340290. Sequence in context: A093504 A370279 A009180 * A041459 A126702 A260126 Adjacent sequences: A359837 A359838 A359839 * A359841 A359842 A359843 KEYWORD nonn,base AUTHOR Bernard Schott, Jan 15 2023 EXTENSIONS More terms from Amiram Eldar, Jan 15 2023 STATUS approved

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Last modified July 22 06:07 EDT 2024. Contains 374481 sequences. (Running on oeis4.)