A069488 -id:A069488 - cp's OEIS frontend

A069489 Primes > 1000 in which every substring of length 3 is also prime.

1013, 1019, 1031, 1097, 1277, 1373, 1499, 1571, 1733, 1811, 1997, 2113, 2239, 2293, 2719, 3079, 3137, 3313, 3373, 3491, 3499, 3593, 3673, 3677, 3733, 3739, 3797, 4013, 4019, 4211, 4337, 4397, 4673, 4877, 4919, 5233, 5419, 5479, 6011, 6073, 6079, 6131

Offset: 1

Amarnath Murthy, Mar 30 2002

Minimum number of digits is taken to be 4 as all 3-digit primes would be trivial members.
Zero may occur only as second digit from left. - Zak Seidov, Dec 28 2020
All the digits after the two first digits from left are necessarily odd. - Bernard Schott, Mar 20 2022

			11317 is a term as the three substrings of length 3 i.e. 113,131 and 317 all are primes.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A069488 and A069490.

Cf. A211685, A010051, A000040.

a069489 n = a069489_list !! (n-1)
a069489_list = filter g $ dropWhile (<= 1000) a000040_list where
   g x = x < 100 || a010051 (x `mod` 1000) == 1 && g (x `div` 10)
-- Reinhard Zumkeller, Apr 07 2014

Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 3, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[1000] + 1, 10^3}]
Select[Prime[Range[169,800]],AllTrue[FromDigits/@Partition[ IntegerDigits[ #],3,1], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 05 2019 *)

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

A069490 Primes > 1000 in which every substring of lengths 2 and 3 are also prime.

Original entry on oeis.org

1373, 3137, 3797, 6131, 6173, 6197, 9719, 11311, 11317, 17971, 31379, 61379, 71971, 113131, 113173, 113797, 131311, 131317, 131797, 179719, 317971, 431311, 431797, 617971, 1131131, 1131379, 1311311, 1313797, 1317971, 3131137, 3131311

Offset: 1

Amarnath Murthy, Mar 30 2002

nonn

For all terms: substrings of length 3 correspond to one of the first 21 terms of A069489. - Reinhard Zumkeller, Jun 08 2015

			11317 is a term as the substrings of length 2, i.e., 11, 13, 31, 17 and the three substrings of length 3, i.e., 113, 131 and 317 are all prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Cf. A069488 and A069489.

Cf. A010051, A038618.

import Data.Set (fromList, deleteFindMin, union)
a069490 n = a069490_list !! (n-1)
a069490_list = f $ fromList [1..9] where
   f s | m < 1000               = f s''
       | h m && a010051' m == 1 = m : f s''
       | otherwise              = f s''
       where s'' = union s' $ fromList $ map (+ (m * 10)) [1, 3, 7, 9]
             (m, s') = deleteFindMin s
   h x = x < 100 && a010051' x == 1 ||
         a010051' (x `mod` 1000) == 1 &&
         a010051' (x `mod` 100) == 1 && h (x `div` 10)
-- Reinhard Zumkeller, Jun 08 2015

Do[ If[ Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 2, 1]]]] == Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[ Prime[n]], 3, 1]]]] == {True}, Print[ Prime[n]]], {n, PrimePi[1000] + 1, 10^5}]
 Select[Prime[Range[169,226000]],AllTrue[FromDigits/@Flatten[Table[ Partition[ IntegerDigits[ #],k,1],{k,{2,3}}],1],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 02 2021 *)

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

A131648 Primes > 100 in which every multi-digit substring is also prime.

Original entry on oeis.org

113, 131, 137, 173, 179, 197, 311, 313, 317, 373, 379, 419, 431, 479, 613, 617, 619, 673, 719, 797, 971, 1373, 3137, 3797, 6131, 6173, 6197, 9719

Offset: 1

Tanya Khovanova, Sep 09 2007

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

			9719 is a member becauase 97, 71, 19, 971, 719, 9719 are all primes.

Cf. This sequence is a subsequence of A069488 - Primes > 100 in which every substring of length 2 is also prime.

Select[Prime[Range[26,1300]],And@@PrimeQ[Flatten[Table[FromDigits/@ Partition[ IntegerDigits[#],n,1],{n,2,IntegerLength[#]-1}]]]&] (* Harvey P. Dale, Nov 03 2012 *)

A226108 Primes remaining prime if all but two digits are deleted.

Original entry on oeis.org

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113, 131, 137, 173, 179, 197, 311, 317, 431, 617, 719, 1117, 1171, 4111, 11113, 11117, 11119, 11131, 11171, 11173, 11197, 11311, 11317, 11719, 11731, 13171, 13711, 41113

Offset: 1

Tim Cieplowski, May 26 2013

Subsequence of A069488.

			For a(3)=137, all pairs of two digits (in their original order) 13, 17, and 37 are prime.

C. Caldwell, Truncatable primes, J. Recreational Math., 19:1 (1987) 30-33.

Tim Cieplowski, Table of n, a(n) for n = 1..2720
P. Ballew, Knockout Primes and a new notation, May 17, 2013
C. Caldwell, The Prime Glossary: Deletable Prime

Cf. A019546, A051362, A069488.

testQ[n_] := n > 9 && Catch[Block[{d = IntegerDigits@n}, Do[If[! PrimeQ[ d[[j]] + 10*d[[i]]], Throw@False], {j, 2, Length@d}, {i, j-1}]; True]]; Select[Prime@ Range[10^5], testQ] (* Giovanni Resta, May 28 2013 *)

A242377 Greatest n-digit prime in which every two-digit string is also a prime.

Original entry on oeis.org

7, 97, 971, 9719, 97973, 979717, 9797371, 97973731, 979797979, 9797979713, 97979797171, 979797973117, 9797979797971, 97979797973173, 979797979797317, 9797979797979719, 97979797979797373, 979797979797971179, 9797979797979737971

Offset: 1

Robert G. Wilson v, May 12 2014

To stay parallel with A090534, a(1) exists.

The greatest potential candidate, gpc(n), is of the form floor(10^n*97/99). The following is gpc(n)-a(n): 2, 0, 8, 78, 6, 80, 608, 6066, 0, 84, 808, 6680, 8, 6624, 662, 78, 606, 8618, 60008, 86, 600, 6000, 660, 608060, 808, ... .

Robert G. Wilson v, Table of n, a(n) for n = 1..100

Cf. A090534, A069488.

fQ[p_] := Block[{id = IntegerDigits@ p}, Union@ PrimeQ[ FromDigits@# & /@ Partition[id, 2, 1]] == {True}]; f[n_] := Block[{p = NextPrime[10^n*97/99, -1]}, While[ !fQ@ p, p = NextPrime[p, -1]]; p]; f[1] = 7; Array[f, 19]

A070024 First prime > 10^n in which every substring of length n is prime.

Original entry on oeis.org

23, 113, 1013, 10139, 100379, 1000037, 10000379, 100000193, 1000001237, 10000000097, 100000000193, 1000000000193, 10000000001777, 100000000001831, 1000000000036931, 10000000000001873, 100000000000000691

Offset: 1

Robert G. Wilson v, Apr 12 2002

Cf. A019546, A069488, A069489, A070025.

Cf. A179335. [From Reinhard Zumkeller, Jul 11 2010]

Do[k = 10^n; While[ !PrimeQ[k] || Union[ PrimeQ[ Map[ FromDigits, Partition[ IntegerDigits[k], n, 1]]]] != {True}, k++ ]; Print[k], {n, 1, 25}]

A211683 Numbers > 100 such that all the substrings of length = 2 are primes.

Original entry on oeis.org

111, 113, 117, 119, 131, 137, 171, 173, 179, 197, 231, 237, 297, 311, 313, 317, 319, 371, 373, 379, 411, 413, 417, 419, 431, 437, 471, 473, 479, 531, 537, 597, 611, 613, 617

Offset: 1

Hieronymus Fischer, Jun 08 2012

Only numbers > 100 are considered, since all 2-digit primes are trivial members. See A069488 for the sequence with prime terms > 100.
The sequence is infinite (for example, consider the continued concatenation of ‘11’ or of ‘13’: 111, 1111, 11111, ..., 131, 1313, 13131, ... are members).
Infinitely many terms are palindromic.

			a(2)=113, since all substrings of length = 2 are primes (11 and 13).
a(10)=197, since all substrings of length = 2 (19, 97) are primes.

Hieronymus Fischer, Table of n, a(n) for n = 1..5000

Cf. A019546, A035232, A039996, A046034, A069488, A085823, A131648, A211681, A211682.

cp's OEIS Frontend

A069489 Primes > 1000 in which every substring of length 3 is also prime.

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A069490 Primes > 1000 in which every substring of lengths 2 and 3 are also prime.

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A131648 Primes > 100 in which every multi-digit substring is also prime.

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A226108 Primes remaining prime if all but two digits are deleted.

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A242377 Greatest n-digit prime in which every two-digit string is also a prime.

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A070024 First prime > 10^n in which every substring of length n is prime.

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Mathematica

A211683 Numbers > 100 such that all the substrings of length = 2 are primes.

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