A087911 -id:A087911 - cp's OEIS frontend

A137779 Number of bases (numbering systems, including unary) in which the n-th prime is a palindrome having at least two digits.

1, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 3, 3, 3, 2, 2, 3, 3, 4, 3, 4, 2, 3, 3, 3, 3, 2, 4, 4, 3, 4, 3, 2, 2, 2, 4, 4, 2, 2, 4, 2, 4, 5, 3, 4, 3, 4, 2, 4, 3, 3, 3, 4, 3, 6, 2, 2, 4, 4, 3, 2, 2, 4, 2, 5, 2, 3, 5, 2, 3, 5, 2, 2, 6, 5, 3, 2, 3, 4, 4, 4, 5, 3, 4, 2, 5, 3, 4, 4, 4, 3, 3, 4, 2, 3, 3, 3, 4, 4

Offset: 1

Attila Olah (jolafix(AT)gmail.com), May 06 2008, corrected May 08 2008

Each prime p > 2 is palindrome in at least base 1 and base p-1, since p = 1*(p-1)^1 + 1*(p-1)^0 and p = 1*1^(p-1) + 1*1(p-2) + ... + 1*1^1 + 1*1^0.

			a(621) = 9 because the 621st prime (4591) is a palindrome in 9 bases: base 1, 19, 20, 24, 33, 37, 51, 54 and 4590 (4591 = 1*4590^1 + 1*4590^0).

Attila Olah, Table of n, a(n) for n = 1..10000

Cf. A087911, A126071.

ispal(v) = {for(i=1, #v\2, if (v[i] != v[#v-i+1], return(0));); return(1);};
a(n) = {p = prime(n); 1 + sum(i=2, p, ispal(digits(p, i)));} \\ Michel Marcus, Sep 04 2013

a(n) = A126071(prime(n)). - Charles R Greathouse IV, Jun 19 2014

A331486 Numbers k which are emirps in more bases 1 < b < k than any previous number.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 53, 67, 73, 79, 97, 113, 127, 157, 193, 223, 269, 277, 311, 379, 421, 431, 467, 487, 563, 613, 647, 743, 907, 937, 977, 1093, 1193, 1249, 1259, 1373, 1483, 1543, 1637, 1667, 1933, 2239, 2393, 2477, 2521, 2857, 2957, 3083

Offset: 1

Amiram Eldar, Jan 23 2020

The corresponding numbers of bases are 0, 1, 3, 6, 8, 9, 12, 13, 17, 21, 24, ... (see the link for more values).

			2 is not emirp in any base.
5 is emirp in one base, 3: 5 is 12 in base 3, and 21 in base 3 is 7 which is also a prime.
7 is emirp in 3 bases, 3, 4, and 5.

Amiram Eldar, Table of n, a(n) for n = 1..175
Amiram Eldar, Table of n, a(n), number of bases for n = 1..175

Cf. A006567, A080790, A087911, A107129, A228768.

emirpQ[n_, b_] := n != (rev = FromDigits[Reverse @ IntegerDigits[n, b], b]) && And @@ PrimeQ[{n, rev}];
emirpCount[n_] := Length @ Select[Range[2, n - 1], emirpQ[n, #] &];
seq = {}; emax = -1; Do[e1 = emirpCount[n]; If[e1 > emax, emax = e1; AppendTo[seq, n]], {n, 2, 3000}]; seq

cp's OEIS Frontend

A137779 Number of bases (numbering systems, including unary) in which the n-th prime is a palindrome having at least two digits.

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