A072885 Primes p with a prime number of digits in all base b with 1 <= b <= p.

2, 3, 5, 7, 17, 19, 23

Offset

1,1

Comments

n in base 1 is understood to be the concatenation of n 1's. Hence a prime in base 1 has a prime number of digits. The (first) seven terms listed also have a prime number of digits when expressed in any negative base b with -p<= b <= -2. Hence they have a noncomposite number of digits for any meaningful base b (for the common use of "base") as only one digit is required whenever b > p or b < -p.

Once you reach the 169th prime, which is 1009, at least for base 10, its representation is of composite length. For any prime above this, at some point in the base representations you must have a base representation that is only four in length.

Links

Table of n, a(n) for n=1..7.

Example

7 is a term because 7 = 1111111 (base 1) = 111 (base 2) = 21 (base 3) = 13 (base 4) = 12 (base 5) = 11 (base6) = 10 (base 7) and the number of digits is 7, 3, or 2, a prime, for each base 1 <= b <=7. Also, in support of the comments, 7 = 11011 (base -2) = 111 (base -3) = 133 (base -4) = 142 (base -5) = 151 (base -6) = 160 (base -7), 3 or 5 digits, again a prime length, for each negative base with -7 <= b = -2.

Mathematica

Do[p = Prime[n]; k = 2; While[k < p && PrimeQ[ Length[ IntegerDigits[p, k]]], k++ ]; If[k == p, Print[p]], {n, 1, 10^6}]

Crossrefs

Sequence in context: A059496 A066814 A358047 * A178209 A360781 A042994

Adjacent sequences: A072882 A072883 A072884 * A072886 A072887 A072888

Keyword

base,nonn,full,fini

Author

Rick L. Shepherd, Jul 28 2002

Extensions

Edited by Robert G. Wilson v, Aug 01 2002

Status

approved