A069488 - OEIS

%I #28 Mar 02 2019 02:48:11

%S 113,131,137,173,179,197,311,313,317,373,379,419,431,479,613,617,619,

%T 673,719,797,971,1117,1171,1319,1373,1973,1979,2311,2371,2971,3119,

%U 3137,3719,3797,4111,4373,6113,6131,6173,6197,6719,6737

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

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

%C From _Robert G. Wilson v_, May 12 2014: (Start)

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

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

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

%C 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.

%C \10^n

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

%C   \

%C 1     0  19   34  146   648  1162  2678   8037  22740   39.188034

%C 2     0   0    3    6    27    18    66    175    449    0.816186

%C 3     0  14   19   63   326   712  1526   3855  11040   19.403018

%C 4     0   3    2   13    54    92   143    384   1031    1.895550

%C 5     0   0    0    9    17    24    45    176    426    0.763995

%C 6     0   4    6    4    24    66   146    233    630    1.224834

%C 7     0  14   20  100   436   907  1980   5442  15421   26.875285

%C 8     0   0    3    6    24    25    37    176    388    0.721797

%C 9     0   9   13   38   157   361   763   1790   5125    9.111301

%C Total 0  63  100  385  1713  3367  7384  20268  57250  100.00000

%C (End)

%H Robert G. Wilson v, <a href="/A069488/b069488.txt">Table of n, a(n) for n = 1..10101</a> (first 1000 terms from Reinhard Zumkeller)

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

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

%o (Haskell)

%o a069488 n = a069488_list !! (n-1)

%o a069488_list = filter f $ dropWhile (<= 100) a038618_list where

%o    f x = x < 10 || a010051 (x `mod` 100) == 1 && f (x `div` 10)

%o -- _Reinhard Zumkeller_, Apr 07 2014

%Y Cf. A069489 and A069490.

%Y Cf. A010051, subsequence of zeroless primes: A038618.

%K nonn,base

%O 1,1

%A _Amarnath Murthy_, Mar 30 2002

%E Edited, corrected and extended by _Robert G. Wilson v_, Apr 12 2002