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

Offset: 1

Author

Thomas Scheuerle, Jan 07 2023

Keywords

• nonn
• base

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.

Examples

			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).
		

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)

Crossrefs

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

Programs

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))

Formula

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