A024770 -id:A024770 - cp's OEIS frontend

A032437 Substrings from the right are prime numbers (using only odd digits different from 5).

3, 7, 13, 17, 37, 73, 97, 113, 137, 173, 197, 313, 317, 337, 373, 397, 773, 797, 937, 997, 1373, 1997, 3137, 3313, 3373, 3797, 7937, 9137, 9173, 9337, 9397, 13313, 33797, 39397, 79337, 79397, 91373, 91997, 99137, 99173, 99397, 139397, 379397

Offset: 1

Carlos Rivera

Primes p with decimal expansion d_1 d_2 d_3 ... d_k such that the digits d_i are 1, 3, 7, or 9, and deleting 1, 2, 3, up to k-1 leading digits also produces a prime. For example, 9173 is a term because all of 9173, 173, 73, and 3 are primes. - N. J. A. Sloane, Jun 28 2022

			173 is a term because 173, 73, and 3 are all primes. 371 is not a term because 371 and 1 are not primes. - _N. J. A. Sloane_, Jun 28 2022

T. D. Noe, Table of n, a(n) for n = 1..58 [The complete list of terms]
C. Rivera, Prime strings
Eric Weisstein's World of Mathematics, Truncatable Prime.

Cf. A033664, A024785, A020994, A024770, A052023, A052024, A052025, A050986, A050987.

Select[Prime[Range[33000]],SubsetQ[{1,3,7,9},IntegerDigits[#]]&&AllTrue[Mod[#,10^Range[ IntegerLength[ #]-1]],PrimeQ]&] (* Harvey P. Dale, Jun 28 2022 *)

is(n)=my(d=digits(n)); for(i=1,n, if(!isprime(fromdigits(d[i..n])), return(0))); 1 \\ Charles R Greathouse IV, Jun 25 2017

Single-digit terms added by Eric W. Weisstein.

A052023 Every suffix of palindromic prime a(n), containing no '0' digit, is prime (left-truncatable palindromic primes).

Original entry on oeis.org

2, 3, 5, 7, 313, 353, 373, 383, 797, 76367, 79397, 7693967, 799636997

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest, Nov 15 1999

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
C. K. Caldwell, Left and Right truncatable primes.
P. De Geest, The list of 4260 left-truncatable primes
Eric Weisstein's World of Mathematics, Prime strings
Index entries for sequences related to truncatable primes

Cf. A033664, A024785, A032437, A020994, A024770, A052024, A052025, A050986, A050987.

d[n_]:=IntegerDigits[n]; ltrQ[n_]:=And@@PrimeQ[NestList[FromDigits[Drop[d[#],1]]&,n,Length[d[n]]-1]]; palQ[n_]:=Reverse[x=d[n]]==x; Select[Prime[Range[540000]],palQ[#]&<rQ[#]&] (* Jayanta Basu, Jun 02 2013 *)

A052024 Every suffix of palindromic prime a(n) is prime (left-truncatable).

Original entry on oeis.org

2, 3, 5, 7, 313, 353, 373, 383, 797, 30103, 31013, 70607, 73037, 76367, 79397, 3002003, 7096907, 7693967, 700090007, 799636997, 70060906007, 3000002000003, 7030000000307, 300000020000003, 300001030100003, 310000060000013, 38000000000000000000083, 30000000004000300040000000003, 3000001000000000000000000000001000003

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest, Nov 15 1999

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
C. K. Caldwell, Left and Right truncatable primes.
Eric Weisstein's World of Mathematics, Prime strings
Index entries for sequences related to truncatable primes

Cf. A002385, A033664, A024785, A032437, A020994, A024770, A052023, A052025, A050986, A050987.

d[n_]:=IntegerDigits[n]; ltrQ[n_]:=And@@PrimeQ[NestWhileList[FromDigits[Drop[d[#],1]]&,n,#>9&]]; palQ[n_]:=Reverse[x=d[n]]==x; Select[Prime[Range[550000]],palQ[#]&<rQ[#]&] (* Jayanta Basu, Jun 02 2013 *)

from sympy import isprime
from itertools import count, islice
def agen(verbose=False):
    prime_strings, alst = {"3", "7"}, []
    yield from [2, 3, 5, 7]
    for digs in count(2):
        new_prime_strings = set()
        for p in prime_strings:
            for d in "123456789":
                ts = d + "0"*(digs-1-len(p)) + p
                if isprime(int(ts)):
                    new_prime_strings.add(ts)
        prime_strings |= new_prime_strings
        pals = [int(s) for s in new_prime_strings if s == s[::-1]]
        yield from sorted(pals)
        if verbose: print("...", digs, len(prime_strings), time()-time0)
print(list(islice(agen(), 20))) # Michael S. Branicky, Apr 04 2022

Inserted missing 31013 by Jayanta Basu, Jun 02 2013

a(27)-a(29) from Michael S. Branicky, Apr 04 2022

A137812 Left- or right-truncatable primes.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 113, 131, 137, 139, 167, 173, 179, 197, 223, 229, 233, 239, 271, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439, 443, 467, 479, 523, 547, 571

Offset: 1

T. D. Noe, Feb 11 2008

Repeatedly removing a digit from either the left or right produces only primes. There are 149677 terms in this sequence, ending with 8939662423123592347173339993799.

The number of n-digit terms is A298048(n). - Jon E. Schoenfield, Jan 28 2022

			139 is here because (removing 9 from the right) 13 is prime and (removing 1 from the left) 3 is prime.

T. D. Noe, Table of n, a(n) for n = 1..10000
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
T. D. Noe, Plot of all terms
Carlos Rivera, Puzzle 2: Prime Strings, The Prime Puzzles and Problems Connection.
Daniel Starodubtsev, Full sequence
Eric Weisstein, MathWorld: Truncatable Prime
Index entries for sequences related to truncatable primes

Cf. A024770 (right-truncatable primes), A024785 (left-truncatable primes), A077390 (left-and-right truncatable primes), A080608.

Cf. A298048 (number of n-digit terms).

Clear[s]; s[0]={2,3,5,7}; n=1; While[s[n]={}; Do[k=s[n-1][[i]]; Do[p=j*10^n+k; If[PrimeQ[p], AppendTo[s[n],p]], {j,9}]; Do[p=10*k+j; If[PrimeQ[p], AppendTo[s[n],p]], {j,9}], {i,Length[s[n-1]]}]; s[n]=Union[s[n]]; Length[s[n]]>0, n++ ];t=s[0]; Do[t=Join[t,s[i]], {i,n}]; t

from sympy import isprime
def agen():
    primes = [2, 3, 5, 7]
    while len(primes) > 0:
        yield from primes
        cands = set(int(d+str(p)) for p in primes for d in "123456789")
        cands |= set(int(str(p)+d) for p in primes for d in "1379")
        primes = sorted(c for c in cands if isprime(c))
afull = [an for an in agen()]
print(afull[:60]) # Michael S. Branicky, Aug 04 2022

A238853 Right-truncatable, reversible primes in base 256.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 773, 809, 823

Offset: 1

Stanislav Sykora, Mar 06 2014

See A238850 for definitions, and A238854 for comments on general context.

In base 256, there are 35127 such numbers (see A238855), shown here in decimal format. Base 256 is of interest to programmers because its digits correspond to 8-bit bytes and are easily readable in hexadecimal.

			The largest such number is 143496996325262301365903209731563 which, written in hex format, with hyphens between bytes for better readability, is 07-13-2F-CD-51-E1-B1-11-EB-23-CD-B3-15-EB. Truncate on the right any number of bytes and the remaining prefix is still a prime, no matter whether the bytes are read from left to right, or vice versa!

Stanislav Sykora, Table of n, a(n) for n = 1..35127
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. All in base 10: A238850, 16: A238851, 100: A238852.

Cf. In base n: A238854 (largest), A238856 (maximum digits), A238857 (m-digits counts). Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

A202259 Right-truncatable nonprimes: every prefix is nonprime number.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 40, 42, 44, 45, 46, 48, 49, 60, 62, 63, 64, 65, 66, 68, 69, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 120, 121, 122, 123, 124, 125, 126, 128, 129, 140, 141, 142

Offset: 1

Jaroslav Krizek, Dec 25 2011

Supersequence of A202260, A202265. - Barry Carter, Sep 16 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Barry Carter, Table of n, a(n) for n=1..23498959 (bz2 compressed)
Barry Carter, Mathematica package

Cf. A012883 (right-truncatable noncomposites), A024770 (right-truncatable primes), A202260 (right-truncatable composites).

filter:= proc(n) option remember; not isprime(n) and procname(floor(n/10)) end proc:
for i from 0 to 9 do filter(i):= not isprime(i) od:
select(filter, [$0..1000]); # Robert Israel, Nov 01 2016

A227919 Primes which remain prime when rightmost digit is removed.

Original entry on oeis.org

23, 29, 31, 37, 53, 59, 71, 73, 79, 113, 131, 137, 139, 173, 179, 191, 193, 197, 199, 233, 239, 293, 311, 313, 317, 373, 379, 419, 431, 433, 439, 479, 593, 599, 613, 617, 619, 673, 677, 719, 733, 739, 797, 839, 971, 977, 1013, 1019, 1031, 1033, 1039, 1091, 1093

Offset: 1

K. D. Bajpai, Oct 10 2013

After a(2), all the terms in this sequence have second digit from right 1, 3, 7, or 9.

			a(7)= 71 which is prime. Removing the rightmost digit gives 7, which is also prime.
a(16)= 191 which is prime. Removing the rightmost digit gives 19, which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040 (prime numbers), A024770 (right-truncatable primes).
Cf. A052025 (palindromic prime is prime right- or left-truncatable).
Cf. A137812 (left- or right-truncatable primes).

KD:= proc() local a,b; a:= ithprime(n); b:=floor(a/10); if isprime(b) then return (a) :fi; end: seq(KD(),n=1..1000);

is(n)=isprime(n) && isprime(n\10) \\ Charles R Greathouse IV, Oct 12 2013

A012883 Numbers in which every prefix (in base 10) is 1 or a prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 53, 59, 71, 73, 79, 113, 131, 137, 139, 173, 179, 191, 193, 197, 199, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 1319, 1373, 1399, 1733, 1913, 1931, 1933, 1973, 1979, 1993, 1997, 1999

Offset: 1

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

T. D. Noe, Table of n, a(n) for n=1..147 (complete sequence)

Cf. A024770 (every prefix is prime).

max = 10^10; truncate[p_] := If[q = Quotient[p, 10]; q == 1 || PrimeQ[q], q, p]; ok[p_] := FixedPoint[truncate, p] < 10; p = 1; cnt = 2; A012883 = Join[{1}, Reap[While[(p = NextPrime[p]) < max, If[ok[p], Print[cnt++, " ", p]; Sow[p]]]][[2, 1]]] (* Jean-François Alcover, Nov 23 2015 *)

for(n=1, 1e4, k=n; while(isprime(n) || n==1, c=n; n=(c-lift(Mod(c, 10)))/10); if(n==0, print1(k, ", ")); n=k) \\ Altug Alkan, Nov 23 2015

Last term is A094335(10) = 1979339339 (confirmed by David W. Wilson ).

A065122 a(1) = 2; a(n) is smallest prime > 10*a(n-1).

Original entry on oeis.org

2, 23, 233, 2333, 23333, 233341, 2333459, 23334601, 233346041, 2333460413, 23334604169, 233346041759, 2333460417637, 23334604176379, 233346041763823, 2333460417638239, 23334604176382489, 233346041763824911

Offset: 1

Henry Bottomley, Nov 13 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A024770, A055496.

NextPrim[n_Integer] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; a[1] = 2; a[n_] := NextPrim[ 10*a[n - 1]]; Table[ a[n], {n, 1, 20} ]
NestList[NextPrime[10#]&,2,20] (* Harvey P. Dale, Jun 03 2012 *)

{ for (n=1, 100, if (n>1, a=nextprime(10*a), a=2); write("b065122.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 10 2009

More terms from Robert G. Wilson v, Nov 28 2001

A238852 Right-truncatable, reversible primes in base 100.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 311, 313, 347, 349, 353, 359, 367, 701, 709, 719, 727, 733, 739, 751, 769, 773, 787, 1103, 1109, 1123, 1163, 1181, 1193, 1301, 1303, 1319, 1321, 1327, 1361, 1777

Offset: 1

Stanislav Sykora, Mar 06 2014

See A238850 for definitions, and A238854 for comments on general context.

In base 100, chosen as one of four examples, there are 1552 such numbers.

			The largest number of this type (using hyphens to separate the base 100 digits) is 19-07-93-27-17-37-99-47-19-11.Truncate any even number of decimal digits on its right, and the remaining prefix is still a base-100 reversible prime (e.g., 19079327 and 27930719 are both primes).

Stanislav Sykora, Table of n, a(n) for n = 1..1552
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. All in base 10: A238850, 16: A238851, 256: A238853.
Cf. In base n: A238854 (largest), A238855 (totals), A238856 (maximum digits), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
See the link.
```

cp's OEIS Frontend

A032437 Substrings from the right are prime numbers (using only odd digits different from 5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A052023 Every suffix of palindromic prime a(n), containing no '0' digit, is prime (left-truncatable palindromic primes).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A052024 Every suffix of palindromic prime a(n) is prime (left-truncatable).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A137812 Left- or right-truncatable primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A238853 Right-truncatable, reversible primes in base 256.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A202259 Right-truncatable nonprimes: every prefix is nonprime number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A227919 Primes which remain prime when rightmost digit is removed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A012883 Numbers in which every prefix (in base 10) is 1 or a prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica