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A069488 Primes > 100 in which every substring of length 2 is also prime. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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)

LINKS

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

EXAMPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A069489 and A069490.

Cf. A010051, subsequence of zeroless primes: A038618.

Sequence in context: A284598 A060591 A214847 * A131648 A180441 A180407

Adjacent sequences:  A069485 A069486 A069487 * A069489 A069490 A069491

KEYWORD

nonn,base

AUTHOR

Amarnath Murthy, Mar 30 2002

EXTENSIONS

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

STATUS

approved

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Last modified August 5 18:00 EDT 2021. Contains 346488 sequences. (Running on oeis4.)