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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.
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%I #40 Feb 13 2023 12:46:19

%S 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,

%T 49,43,52,37,60,65,53,59,57,50,52,85,52,79,76,57,77,69,103,90,83,84,

%U 106,80,68,90,85,89,94,75,100,108,87,128,97,119,99,118,139,105,96

%N 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.

%C 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.

%C If a(n) is a record in this sequence, then A000040(n) is in A331486.

%C 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.

%H Thomas Scheuerle, <a href="/A359113/b359113.txt">Table of n, a(n) for n = 1..4000</a>

%H Thomas Scheuerle, <a href="/A359113/a359113.png">Scatterplot a(n)/(n-1) for the first 2000 values</a>. 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.

%H Rémy Sigrist, <a href="/A359113/a359113_1.png">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)</a>

%F a(n) >= A135551(A000040(n)).

%e a(3) = 3:

%e prime(3) = 5 in bases 2..5:

%e 5 = 101_2; reversing digits gives 101_2 = 5 (prime).

%e 5 = 12_3; reversing digits gives 21_3 = 7 (prime).

%e 5 = 11_4; reversing digits gives 11_4 = 5 (prime).

%e 5 = 10_5; reversing digits gives 01_5 = 1 (nonprime).

%o (PARI) revprime(p, b)=my(q, t=p); while(t, q=b*q+t%b; t\=b); isprime(q)

%o a(n) = sum(b = 2, prime(n), revprime(prime(n), b))

%Y Cf. A000040, A007500, A002385, A135551, A228768, A331486.

%K nonn,base

%O 1,3

%A _Thomas Scheuerle_, Jan 07 2023