A068669 -id:A068669 - cp's OEIS frontend

A085823 Numbers in which all substrings are primes.

2, 3, 5, 7, 23, 37, 53, 73, 373

Offset: 1

Zak Seidov, Jul 04 2003

The definition implies that the number itself must be prime.
Apparently there are no such primes > 373.
From Jean-Marc Falcoz, Jan 11 2009: (Start)
This is correct.
There can't be any more terms, because they must necessarily be of the form
23737373733737... but the substring 237 is composite
or 273737373... but 273 is composite
or 5373737373... but 537 is composite
or 5737373737... but 573 is composite
or 37373737373... but 3737 is composite
or 7373737373... but 737 is composite
No other form is possible, otherwise, if the digit 2 or 5 is anywhere inside or at the end of the number, one substring-number is even or divisible by 5, and furthermore, there can't be twin digits, because one substring-number would then be divisible by 11.
Obviously, the digits 0, 1, 4, 6, 8, 9 can't appear anywhere in a term of the sequence. (End)
Subsequence of A024770 (right-truncatable primes), A068669 (noncomposite numbers in which all substrings are noncomposite). Supersequence of A202263 (primes in which all substrings and reversal substrings are primes). - Jaroslav Krizek, Jan 28 2012.
From Hieronymus Fischer, Apr 20 2012: (Start)
A more general definition is "Numbers such that all substrings of length <= 3 are primes". Proof: For numbers < 1000 this is plainly true. Suppose that there are such n >= 1000. We recognize that n must contain the string 373, as this is the only valid prime substring with the length 3. It follows, that there are substrings x37 or 73x, with any digit x. Evidently, neither x37 nor 73x are valid prime substrings, independent from the digit x. Thus, there is no number >= 1000 such that all substrings of length <= 3 are primes.
Also, numbers such that all substrings of length <= 2 are primes and the number of prime substrings of length = 3 is greater than m-3 for n <= 37373 and is greater than min(m-4,floor((m-1)/2) else; where m=floor(log_10(a(n)))+1 = number of digits. (End)

			Example : 373 is in the sequence, because 3, 7, 37, 73 and 373 are prime, but 733 is not in the sequence, because 33 is not prime.

Onno M. Cain, Prime-bounded subwords, arXiv:1912.08598 [math.HO], 2019.
NRICH, Strange Numbers
Henri Picciotto's Math Education Page, "Super-slimes" in Infinity, Unit 1

Cf. A085822, A166504, A213300.

Select[Prime@ Range[10^3], AllTrue[FromDigits /@ Rest@ Subsequences@ IntegerDigits@ #, PrimeQ] &] (* Michael De Vlieger, Jul 30 2018 *)

Thanks to Mark Underwood for pointing out misprints in the first version of this sequence.

Edited by N. J. A. Sloane, Jun 20 2009 at the suggestion of Lekraj Beedassy

A024770 Right-truncatable primes: every prefix is prime.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, 23333, 23339, 23399, 23993, 29399, 31193

Offset: 1

David W. Wilson

Primes in which repeatedly deleting the least significant digit gives a prime at every step until a single-digit prime remains. The sequence ends at a(83) = 73939133 = A023107(10).
The subsequence which consists of the following "chain" of consecutive right truncatable primes: 73939133, 7393913, 739391, 73939, 7393, 739, 73, 7 yields the largest sum, compared with other chains formed from subsets of this sequence: 73939133 + 7393913 + 739391 + 73939 + 7393 + 739 + 73 + 7 = 82154588. - Alexander R. Povolotsky, Jan 22 2008
Can also be seen as a table whose n-th row lists the n-digit terms; row lengths (0 for n >= 9) are given by A050986. The sequence can be constructed starting with the single-digit primes and appending, for each p in the list, the primes within 10*p and 10(p+1), formed by appending a digit to p. - M. F. Hasler, Nov 07 2018

Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer London 2010, pp. 86-89.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 112-113.

Jens Kruse Andersen, Table of n, a(n) for n = 1..83 (The full list of terms, taken from link below)
Jens Kruse Andersen, Right-truncatable primes
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Patrick De Geest, The list of 4260 left-truncatable primes
R. Schroeppel, HAKMEM item 33; "Russian Doll Primes", but with a slightly different definition.
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Supersequence of A085823, A202263. Subsequence of A012883, A068669. - Jaroslav Krizek, Jan 28 2012
Supersequence of A239747.
Cf. A033664, A024785 (left-truncatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866, A077390 (left-and-right-truncatable primes), A137812 (left-or-right truncatable primes), A254751, A254753.
Cf. A237600 for the base-16 analog.

import Data.List (inits)
a024770 n = a024770_list !! (n-1)
a024770_list = filter (\x ->
   all (== 1) $ map (a010051 . read) $ tail $ inits $ show x) a038618_list
-- Reinhard Zumkeller, Nov 01 2011

s:=[1,3,7,9]: a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: do for j from l1 to l2 do for k from 1 to 4 do d:=[s[k],op(a[j])]: if(isprime(op(convert(d, base, 10, 10^nops(d)))))then a:=[op(a), d]: fi: od: od: l1:=l2+1: l2:=nops(a): if(l1>l2)then break: fi: od: seq(op(convert(a[j], base, 10, 10^nops(a[j]))),j=1..nops(a)); # Nathaniel Johnston, Jun 21 2011

max = 100000; truncate[p_] := If[PrimeQ[q = Quotient[p, 10]], q, p]; ok[p_] := FixedPoint[ truncate, p] < 10; p = 1; A024770 = {}; While[ (p = NextPrime[p]) < max, If[ok[p], AppendTo[ A024770, p]]]; A024770 (* Jean-François Alcover, Nov 09 2011, after Pari *)
eppQ[n_]:=AllTrue[FromDigits/@Table[Take[IntegerDigits[n],i],{i, IntegerLength[ n]-1}], PrimeQ]; Select[Prime[Range[3400]],eppQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 14 2015 *)

{fileO="b024770.txt";v=vector(100);v[1]=2;v[2]=3;v[3]=5;v[4]=7;j=4;j1=1; write(fileO,"1 2");write(fileO,"2 3");write(fileO,"3 5");write(fileO,"4 7"); until(0,if(j1>j,break);new=1;for(i=j1,j,if(new,j1=j+1;new=0);for(k=1,9, z=10*v[i]+k;if(isprime(z),j++;v[j]=z;write(fileO,j," ",z);))));} \\ Harry J. Smith, Sep 20 2008

for(n=2, 31193, v=n; while(isprime(n), c=n; n=(c-lift(Mod(c, 10)))/10); if(n==0, print1(v, ", ")); n=v); \\ Arkadiusz Wesolowski, Mar 20 2014

A024770=vector(9, n, p=concat(apply(t->primes([t, t+1]*10), if(n>1, p)))) \\ The list of n-digit terms, 1 <= n <= 9. Use concat(%) to "flatten" it. - M. F. Hasler, Nov 07 2018

from sympy import primerange
p = lambda x: list(primerange(x, x+10)); A024770 = p(0); i=0
while iA024770): A024770+=p(A024770[i]*10); i+=1 # M. F. Hasler, Mar 11 2020

A202262 Composite numbers in which all substrings are composite.

Original entry on oeis.org

4, 6, 8, 9, 44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 444, 446, 448, 464, 466, 468, 469, 484, 486, 488, 494, 496, 498, 644, 646, 648, 649, 664, 666, 668, 669, 684, 686, 688, 694, 696, 698, 699, 844, 846, 848, 849, 864, 866, 868, 869, 884, 886

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A062115, A202260, A029581.
Supersequence of A202265.
This is a 10-automatic sequence, see A071062. - Charles R Greathouse IV, Jan 01 2012

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A202262
Index entries for 10-automatic sequences

Cf. A085823 (primes in which all substrings are primes), A068669 (noncomposite numbers in which all substrings are noncomposite), A062115 (nonprimes in which all substrings are nonprimes).

Cf. A010051.

sub[n_] := Block[{d = IntegerDigits[n]}, Union@ Reap[ Do[Sow@ FromDigits@ Take[d, {i, j}], {j, Length@ d}, {i, j}]][[2, 1]]]; Select[ Range@ 900, Union[{4, 6, 8, 9}, IntegerDigits[#]] == {4, 6, 8, 9} && AllTrue[sub[#], CompositeQ] &] (* Giovanni Resta, Dec 20 2019 *)

PARI
```
See Links section.
```

Data corrected by Reinhard Zumkeller, May 05 2012
Data corrected by Rémy Sigrist, Dec 19 2019
Incorrect Haskell program deleted by M. F. Hasler, Dec 20 2019

A202265 Nonprime numbers in which all substrings and reversal substrings are nonprimes.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 18, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 81, 84, 86, 88, 90, 94, 96, 99, 100, 108, 180, 184, 186, 400, 404, 406, 408, 440, 444, 446, 448, 460, 464, 466, 468, 469, 480, 481, 484, 486, 488, 490, 494, 496, 600, 604, 606, 608, 609

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A062115, A202259.

Supersequence of A202266.

			All substrings and reversal substrings of 186 are nonprimes: 1, 6, 8, 18, 68, 81, 86, 186, 681.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A202263 (primes in which all substrings and reversal substrings are primes), A068669 (noncomposite numbers in which all substrings and reversal substrings are noncomposites), A202266 (composite numbers in which all substrings and reversal substrings are composites).

Select[Range[0,1000],NoneTrue[Union[Flatten[{#,IntegerReverse[#]}&/@Flatten[Table[ FromDigits/@Partition[IntegerDigits[#],d,1],{d,IntegerLength[#]}]]]],PrimeQ]&] (* Harvey P. Dale, Jul 30 2024 *)

Corrected (498 deleted) by Harvey P. Dale, Jul 30 2024

A012884 Numbers such that every prefix and suffix is 1 or a prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 197, 311, 313, 317, 373, 797, 1373, 1997, 3137, 3797, 7331, 73331, 739397

Offset: 1

Larry Calmer (larry(AT)wri.com), Simon Plouffe

Last term is 739397 (confirmed by David W. Wilson).

Intersection of A012883 and A143390. [Reinhard Zumkeller, Aug 13 2008]

Cf. A068669.

prQ[n_] := n == 1 || PrimeQ[n];
okQ[n_] := Module[{dd, nd}, dd = IntegerDigits[n]; nd = Length[dd]; AllTrue[Range[nd], prQ@ FromDigits@ Take[dd, #]&] && AllTrue[Range[nd-1], prQ@ FromDigits@ Drop[dd, #]&]];
{1}~Join~Select[Prime@Range[60000], okQ] (* Jean-François Alcover, Nov 20 2019 *)

A202263 Primes in which all substrings and reversal substrings are primes.

Original entry on oeis.org

2, 3, 5, 7, 37, 73, 373

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 7 terms.

Subsequence of A085823, A068669, A024770, A012883.

			All substrings and reversal substrings of 373 are primes:3, 7, 37, 73, 373.

Cf. A202264 (noncomposite numbers in which all substrings and reversal substrings are noncomposite), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

A202264 Noncomposite numbers in which all substrings and reversal substrings are noncomposites.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 113, 131, 311, 313, 373

Offset: 1

Jaroslav Krizek, Dec 25 2011

Sequence is finite with 17 terms.
Supersequence of A202263, A085823.
Subsequence of A068669, A012883, A024770, A012883.

			All substrings and reversal substrings of 311 are noncomposites: 1, 3, 11, 13, 31, 113, 311.

Cf. A202263 (primes in which all substrings and reversal substrings are primes), A202265 (nonprimes in which all substrings and reversal substrings are nonprimes), A202266 (composite numbers in which all substrings and reversal substrings are composites).

A316412 Positive numbers m so that deletion of some or none but not all digits from m yields a noncomposite number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 53, 71, 73, 113, 131, 137, 173, 311, 317

Offset: 1

Matej Kripner, Aug 04 2018

Subsequence of A068669. It is easy to see that these are the only terms from the said sequence that satisfy our definition; there are no more terms < 10000. If there is one >= 10000 then there would be one in [1000, 9999]. A contradiction hence the sequence is finite and full.

Also noncomposites m (in base 10) for which the concatenation of every subsequence of digits of m is noncomposite (in base 10). - David A. Corneth, Aug 08 2018

			317 is a member since all its subsequences, i.e., 3, 1, 7, 31, 17, 37, 317, are noncomposite.
313 is not a member since one of its subsequences (33) is composite.

Subsequence of A068669.

Cf. A008578.

Select[Range[10^3], AllTrue[FromDigits /@ Union@ Rest@ Subsets@ IntegerDigits@ #, ! CompositeQ@ # &] &] (* Michael De Vlieger, Aug 05 2018 *)

A366826 Composite numbers whose proper substrings (of their decimal expansions) are all primes.

Original entry on oeis.org

4, 6, 8, 9, 22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 537, 737

Offset: 1

Kalle Siukola, Oct 25 2023

There are no terms greater than 999 because the only three-digit prime whose substrings are all primes is 373 (see A085823) and prepending or appending any prime digit to it would create a different three-digit substring.

			237 is included because it is composite and 2, 3, 7, 23 and 37 are all primes.
4 is included because it is composite and has no proper substrings.

Subsequence of A002808.
Cf. A000040.
Cf. A061371, A062115, A202262.
Cf. A033274, A068669, A085823, A279366.

from itertools import combinations
from sympy import isprime
for n in range(2, 1000):
    if not isprime(n):
        properSubstrings = set(
            int(str(n)[start:end]) for (start, end)
            in combinations(range(len(str(n)) + 1), 2)
        ) - set((n,))
        if all(isprime(s) for s in properSubstrings):
            print(n, end=', ')

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A085823 Numbers in which all substrings are primes.

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A024770 Right-truncatable primes: every prefix is prime.

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A202262 Composite numbers in which all substrings are composite.

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A202265 Nonprime numbers in which all substrings and reversal substrings are nonprimes.

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A012884 Numbers such that every prefix and suffix is 1 or a prime.

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A202263 Primes in which all substrings and reversal substrings are primes.

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A202264 Noncomposite numbers in which all substrings and reversal substrings are noncomposites.

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A316412 Positive numbers m so that deletion of some or none but not all digits from m yields a noncomposite number.

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