A359113 - OEIS

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.

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)