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 A348226 a(n) is the smallest positive integer that when expressed in bases 2 to n, but read in base n, is always prime. 1
 2, 2, 43, 2, 45481, 2, 65484343, 186914543201, 50006393431, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS a(n)=2 whenever n is prime. Proof: Let n be a prime number. 2 expressed in any base larger than 2 is still 2, which is prime. 2 expressed in base 2 is 10. And 10 read in base n is 1*n + 0 = n, which is prime. The sequence, even when prime indexes are omitted, is not necessarily increasing. Proof: a(9) > a(10). LINKS Table of n, a(n) for n=2..11. EXAMPLE a(4) = 43, because 43 is prime 43 in base 3 is 1121 = 1*3^3 + 1*3^2 + 2*3 + 1 and 1*4^3 + 1*4^2 + 2*4 + 1 = 89, which is prime; 43 in base 2 is 101011 = 1*2^5 + 0*2^4 + 1*2^3 + 0*2^2 + 1*2^1 + 1 and 1*4^5 + 0*4^4 + 1*4^3 + 0*4^2 + 1*4^1 + 1 = 1093, which is prime; and 43 is the smallest positive integer with this property. a(10) = 50006393431 = 153060758677_9 = 564447201127_8 = 3420130221331_7 = 34550030320411_6 = 1304403114042211_5 = 232210213100021113_4 = 11210002000211222202121_3 = 101110100100100111010000001001010111_2; if we read these numbers as base-10 numbers, they are all prime. And 50006393431 is the smallest positive integer with this property. PROG (PARI) isok(k, n) = {for (b=2, n, if (! ispseudoprime(fromdigits(digits(k, b), n)), return (0)); ); return (1); } a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Oct 09 2021 (Python) from gmpy2 import digits, is_prime, next_prime def A348226(n): # code assumes n <= 63 or n is prime if is_prime(n): return 2 p = 2 while True: for i in range(n-1, 1, -1): s = digits(p, i) if not is_prime(int(s, n)): break else: return p p = next_prime(p) # Chai Wah Wu, Nov 19 2021 CROSSREFS Sequence in context: A152016 A264516 A271845 * A280061 A349602 A156467 Adjacent sequences: A348223 A348224 A348225 * A348227 A348228 A348229 KEYWORD nonn,base,more AUTHOR Jesús Bellver Arnau, Oct 09 2021 STATUS approved

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Last modified September 22 18:47 EDT 2023. Contains 365531 sequences. (Running on oeis4.)