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 A359113 a(n) counts the bases b in the interval 2 to p = prime(n), where p if written in base b gives again a prime number in base b if all digits are written in reverse order. 1
 0, 1, 3, 5, 7, 10, 12, 9, 14, 15, 20, 19, 23, 26, 24, 33, 22, 30, 38, 36, 40, 39, 38, 33, 54, 49, 43, 52, 37, 60, 65, 53, 59, 57, 50, 52, 85, 52, 79, 76, 57, 77, 69, 103, 90, 83, 84, 106, 80, 68, 90, 85, 89, 94, 75, 100, 108, 87, 128, 97, 119, 99, 118, 139, 105, 96 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Let p' be p with digit reversal in base b. If p' is composite then all multiplication operations c * d = p' in base b of integers c,d > 1 are using carry in long multiplication. For A000040(n) this is the case in A000040(n) - (a(n)+1) bases. If a(n) is a record in this sequence, then A000040(n) is in A331486. Prime indices of numbers in A228768 are also among the indices of the records in the rational number sequence a(n)/(n-1) with n > 1. See also the plot of this sequence in the link section. LINKS Thomas Scheuerle, Table of n, a(n) for n = 1..4000 Thomas Scheuerle, Scatterplot a(n)/(n-1) for the first 2000 values. Will this quotient remain inside the interval 1..2.5 for any n? For n = 2..2000 a mean value of approximately 1.65... (orange line in plot) was observed. Rémy Sigrist, Scatterplot of (n, b) such that the reversal of prime(n) in base b gives a prime number with n, b <= 2000 and 2 <= b <= prime(n) FORMULA a(n) >= A135551(A000040(n)). EXAMPLE a(3) = 3: prime(3) = 5 in bases 2..5: 5 = 101_2; reversing digits gives 101_2 = 5 (prime). 5 = 12_3; reversing digits gives 21_3 = 7 (prime). 5 = 11_4; reversing digits gives 11_4 = 5 (prime). 5 = 10_5; reversing digits gives 01_5 = 1 (nonprime). PROG (PARI) revprime(p, b)=my(q, t=p); while(t, q=b*q+t%b; t\=b); isprime(q) a(n) = sum(b = 2, prime(n), revprime(prime(n), b)) CROSSREFS Cf. A000040, A007500, A002385, A135551, A228768, A331486. Sequence in context: A206812 A189372 A108052 * A046868 A225240 A104309 Adjacent sequences: A359110 A359111 A359112 * A359114 A359115 A359116 KEYWORD nonn,base AUTHOR Thomas Scheuerle, Jan 07 2023 STATUS approved

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Last modified September 28 13:18 EDT 2023. Contains 365735 sequences. (Running on oeis4.)