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

%I A069488 #28 Mar 02 2019 02:48:11
%S A069488 113,131,137,173,179,197,311,313,317,373,379,419,431,479,613,617,619,
%T A069488 673,719,797,971,1117,1171,1319,1373,1973,1979,2311,2371,2971,3119,
%U A069488 3137,3719,3797,4111,4373,6113,6131,6173,6197,6719,6737
%N A069488 Primes > 100 in which every substring of length 2 is also prime.
%C A069488 Minimum number of digits is taken to be 3 as all two-digit primes would be trivial members.
%C A069488 From _Robert G. Wilson v_, May 12 2014: (Start)
%C A069488 The number of terms below 10^n: 0, 0, 21, 46, 123, 329, 810, 1733, 3985, 9710, ..., .
%C A069488 The least term with n digits is:  113, 1117, 11113, 111119, ..., see A090534.
%C A069488 The largest term with n digits is: 971, 9719, 97973, 979717, ..., see A242377.
%C A069488 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 A069488 \10^n
%C A069488 d\  1&2   3    4    5     6     7     8      9     10 Total % @ 10^10
%C A069488   \
%C A069488 1     0  19   34  146   648  1162  2678   8037  22740   39.188034
%C A069488 2     0   0    3    6    27    18    66    175    449    0.816186
%C A069488 3     0  14   19   63   326   712  1526   3855  11040   19.403018
%C A069488 4     0   3    2   13    54    92   143    384   1031    1.895550
%C A069488 5     0   0    0    9    17    24    45    176    426    0.763995
%C A069488 6     0   4    6    4    24    66   146    233    630    1.224834
%C A069488 7     0  14   20  100   436   907  1980   5442  15421   26.875285
%C A069488 8     0   0    3    6    24    25    37    176    388    0.721797
%C A069488 9     0   9   13   38   157   361   763   1790   5125    9.111301
%C A069488 Total 0  63  100  385  1713  3367  7384  20268  57250  100.00000
%C A069488 (End)
%H A069488 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 A069488 3719 is a term as the three substrings of length 2, i.e., 37, 71 and 19, are all prime.
%t A069488 Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 2, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[100] + 1, 500}]
%o A069488 (Haskell)
%o A069488 a069488 n = a069488_list !! (n-1)
%o A069488 a069488_list = filter f $ dropWhile (<= 100) a038618_list where
%o A069488    f x = x < 10 || a010051 (x `mod` 100) == 1 && f (x `div` 10)
%o A069488 -- _Reinhard Zumkeller_, Apr 07 2014
%Y A069488 Cf. A069489 and A069490.
%Y A069488 Cf. A010051, subsequence of zeroless primes: A038618.
%K A069488 nonn,base
%O A069488 1,1
%A A069488 _Amarnath Murthy_, Mar 30 2002
%E A069488 Edited, corrected and extended by _Robert G. Wilson v_, Apr 12 2002