A069488 Primes > 100 in which every substring of length 2 is also prime.

113, 131, 137, 173, 179, 197, 311, 313, 317, 373, 379, 419, 431, 479, 613, 617, 619, 673, 719, 797, 971, 1117, 1171, 1319, 1373, 1973, 1979, 2311, 2371, 2971, 3119, 3137, 3719, 3797, 4111, 4373, 6113, 6131, 6173, 6197, 6719, 6737

Offset: 1

Author

Amarnath Murthy, Mar 30 2002

Keywords

• nonn
• base

Comments

Minimum number of digits is taken to be 3 as all two-digit primes would be trivial members.

From Robert G. Wilson v, May 12 2014: (Start)

The number of terms below 10^n: 0, 0, 21, 46, 123, 329, 810, 1733, 3985, 9710, ..., .

The least term with n digits is: 113, 1117, 11113, 111119, ..., see A090534.

The largest term with n digits is: 971, 9719, 97973, 979717, ..., see A242377.

The digits 2, 4, 5, 6 and 8 can only appear at the beginning of the prime and the digit 0 never appears. But the digits 1, 3, 7 and 9 can appear anywhere, yet only 1,1 can appear as a pair.

\10^n

d\ 1&2 3 4 5 6 7 8 9 10 Total % @ 10^10

\

1 0 19 34 146 648 1162 2678 8037 22740 39.188034

2 0 0 3 6 27 18 66 175 449 0.816186

3 0 14 19 63 326 712 1526 3855 11040 19.403018

4 0 3 2 13 54 92 143 384 1031 1.895550

5 0 0 0 9 17 24 45 176 426 0.763995

6 0 4 6 4 24 66 146 233 630 1.224834

7 0 14 20 100 436 907 1980 5442 15421 26.875285

8 0 0 3 6 24 25 37 176 388 0.721797

9 0 9 13 38 157 361 763 1790 5125 9.111301

Total 0 63 100 385 1713 3367 7384 20268 57250 100.00000

(End)

Examples

			3719 is a term as the three substrings of length 2, i.e., 37, 71 and 19, are all prime.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10101 (first 1000 terms from Reinhard Zumkeller)

Crossrefs

Cf. A069489 and A069490.

Cf. A010051, subsequence of zeroless primes: A038618.

Programs

Haskell

a069488 n = a069488_list !! (n-1)
a069488_list = filter f $ dropWhile (<= 100) a038618_list where
   f x = x < 10 || a010051 (x `mod` 100) == 1 && f (x `div` 10)
-- Reinhard Zumkeller, Apr 07 2014

Mathematica

Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 2, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[100] + 1, 500}]

Extensions

Edited, corrected and extended by Robert G. Wilson v, Apr 12 2002