A024785 -id:A024785 - cp's OEIS frontend

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

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

A227916 Primes that remain prime when the leftmost digit is removed.

Original entry on oeis.org

13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 131, 137, 167, 173, 179, 197, 211, 223, 229, 241, 271, 283, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 419, 431, 443, 461, 467, 479, 503, 523, 541, 547, 571, 607, 613, 617

Offset: 1

K. D. Bajpai, Oct 13 2013

			a(11)= 97 which is prime. Removing the leftmost digit gives 7, also prime.
a(28)= 311 which is prime. Removing the leftmost digit gives 11, also prime.

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

Cf. A000040 (prime numbers), A024785 (left-truncatable primes).
Cf. A137812 (left- or right-truncatable primes).
Cf. A227919 (primes which remain prime when rightmost digit is removed).

KD:= proc() local a,b,c,d; a:=ithprime(n);b:=length(a); c:=floor(a/(10^(b-1)));d:=a-c*(10^(b-1));if isprime(d) then return(a):fi; end:seq(KD(),n=1..5000);

A024781 Every suffix prime and no 0 digits in base 6 (written in base 6).

Original entry on oeis.org

2, 3, 5, 15, 25, 35, 45, 115, 125, 135, 215, 225, 245, 335, 345, 435, 445, 515, 525, 1115, 1125, 1245, 1335, 1345, 1435, 1445, 2115, 2135, 2225, 2335, 2345, 2435, 3125, 3445, 3515, 4115, 4215, 4225, 4435, 4525, 5215, 5245, 5345, 5525, 11115, 11245, 12135

Offset: 1

David W. Wilson

The final term is a(454) = 14141511414451435.

Nathaniel Johnston, Table of n, a(n) for n = 1..454 (full sequence)

Cf. A024779, A024780, A024781, A024782, A024783, A024784, A024785

a:=[[2], [3], [5]]: b:=[]: l1:=1: l2:=5: do for j from l1 to l2 do for k from 1 to 5 do d:=[op(a[j]),k]: if(isprime(op(convert(d, base, 6, 6^nops(d)))))then a:=[op(a), d]: fi: od: od: l1:=l2+1: l2:=nops(a): if(l1>l2)then break: fi: od: for j from 1 to nops(a) do b:=[op(b),op(convert(a[j], base, 10, 10^nops(a[j])))]: od: b:=sort(b): seq(b[j],j=1..nops(b)); # Nathaniel Johnston, Jun 21 2011

from sympy import isprime
def afull():
    prime_strings, alst = list("235"), []
    while len(prime_strings) > 0:
        alst.extend(sorted(int(p) for p in prime_strings))
        candidates = set(d+p for p in prime_strings for d in "12345")
        prime_strings = [c for c in candidates if isprime(int(c, 6))]
    return alst
print(afull()) # Michael S. Branicky, Apr 27 2022

A024783 Every suffix prime and no 0 digits in base 8 (written in base 8).

Original entry on oeis.org

2, 3, 5, 7, 13, 15, 23, 27, 35, 37, 45, 53, 57, 65, 73, 75, 123, 145, 153, 213, 227, 235, 265, 323, 337, 345, 357, 373, 415, 445, 475, 513, 535, 557, 565, 573, 615, 645, 657, 673, 715, 723, 737, 753, 775, 1145, 1153, 1357, 1475, 1737, 1775, 2213, 2235, 2535, 3123, 3145

Offset: 1

David W. Wilson

The final term of the sequence is a(446) = 313636165537775.

Nathaniel Johnston, Table of n, a(n) for n = 1..446 (full sequence)

Cf. A024779, A024780, A024781, A024782, A024784, A024785.

a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: do for k from 1 to 7 do for j from l1 to l2 do d:=[op(a[j]),k]: if(isprime(op(convert(d, base, 8, 8^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

from sympy import isprime
def afull():
    prime_strings, alst = list("2357"), []
    while len(prime_strings) > 0:
        alst.extend(sorted(int(p) for p in prime_strings))
        candidates = set(d+p for p in prime_strings for d in "1234567")
        prime_strings = [c for c in candidates if isprime(int(c, 8))]
    return alst
print(afull()) # Michael S. Branicky, Apr 27 2022

A024784 Every suffix prime and no 0 digits in base 9 (written in base 9).

Original entry on oeis.org

2, 3, 5, 7, 12, 25, 32, 45, 47, 52, 65, 67, 87, 212, 232, 267, 287, 425, 432, 447, 465, 625, 667, 812, 832, 847, 867, 887, 2287, 2432, 4212, 4465, 4667, 4832, 4847, 4887, 6212, 6267, 6425, 6832, 6887, 8287, 8447, 8667, 8812, 22287, 24465, 24847, 26212, 26887

Offset: 1

David W. Wilson

The final term of the sequence is a(108) = 4284484465.

Nathaniel Johnston, Table of n, a(n) for n = 1..108 (full sequence)

Cf. A024779, A024780, A024781, A024782, A024783, A024785.

a:=[[2],[3],[5],[7]]: l1:=1: l2:=4: do for k from 1 to 8 do for j from l1 to l2 do d:=[op(a[j]),k]: if(isprime(op(convert(d, base, 9, 9^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

from sympy import isprime
def afull():
    prime9strings, alst = list("2357"), []
    while len(prime9strings) > 0:
        alst.extend(sorted(int(p) for p in prime9strings))
        candidates = set(d+p for p in prime9strings for d in "12345678")
        prime9strings = [c for c in candidates if isprime(int(c, 9))]
    return alst
print(afull()) # Michael S. Branicky, Apr 27 2022

A024779 Every suffix is prime and contains no 0 digits in base 4 (written in base 4).

Original entry on oeis.org

2, 3, 13, 23, 113, 223, 323, 1223, 2113, 3323, 21223, 32113, 33323, 233323, 321223, 333323

Offset: 1

David W. Wilson

Cf. A024780, A024781, A024782, A024783, A024784, A024785.

from sympy import isprime
def afull():
    prime_strings, alst = list("23"), []
    while len(prime_strings) > 0:
        alst.extend(sorted(int(p) for p in prime_strings))
        candidates = set(d+p for p in prime_strings for d in "123")
        prime_strings = [c for c in candidates if isprime(int(c, 4))]
    return alst
print(afull()) # Michael S. Branicky, Apr 27 2022

A024780 Every suffix prime and no 0 digits in base 5 (written in base 5).

Original entry on oeis.org

2, 3, 12, 23, 32, 43, 232, 243, 412, 423, 2232, 4412, 4423, 22232, 222232

Offset: 1

David W. Wilson

Cf. A024779, A024781, A024782, A024783, A024784, A024785.

from sympy import isprime
def afull():
    prime_strings, alst = list("23"), []
    while len(prime_strings) > 0:
        alst.extend(sorted(int(p) for p in prime_strings))
        candidates = set(d+p for p in prime_strings for d in "1234")
        prime_strings = [c for c in candidates if isprime(int(c, 5))]
    return alst
print(afull()) # Michael S. Branicky, Apr 27 2022

A024782 Every suffix prime and no 0 digits in base 7 (written in base 7).

Original entry on oeis.org

2, 3, 5, 23, 25, 32, 43, 52, 65, 443, 452, 623, 625, 632, 652, 2452, 2623, 6625, 6652, 42623, 642623, 6642623

Offset: 1

David W. Wilson

Cf. A024779, A024780, A024781, A024783, A024784, A024785.

a:=[[2], [3], [5]]: b:=[]: l1:=1: l2:=3: do for k from 1 to 6 do for j from l1 to l2 do d:=[op(a[j]),k]: if(isprime(op(convert(d, base, 7, 7^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

from sympy import isprime
def afull():
    prime_strings, alst = list("235"), []
    while len(prime_strings) > 0:
        alst.extend(sorted(int(p) for p in prime_strings))
        candidates = set(d+p for p in prime_strings for d in "123456")
        prime_strings = [c for c in candidates if isprime(int(c, 7))]
    return alst
print(afull()) # Michael S. Branicky, Apr 27 2022

A077390 Primes which leave primes at every step if most significant digit and least significant digit are deleted until a one digit or two digit prime is obtained.

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, 127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659

Offset: 1

Amarnath Murthy, Nov 07 2002

There are exactly 920720315 such primes, the largest being 9161759674286961988443272139114537477768682563429152377117139 1111313737919133977331737137933773713713973. - Karl W. Heuer, Apr 19 2011
There are exactly 331780864 odd length primes and 588939451 even length primes, the largest odd length prime being
7228828176786792552781668926755667258635743361825711373791931117197999133917737137399993737111177. - Seth A. Troisi, May 07 2019
Zeros are not permitted, otherwise this sequence would potentially be infinite (cf. A077391). - Sean A. Irvine, May 19 2025

			21313 is a member as 21313, 131 and 3 all are primes.

T. D. Noe, Table of n, a(n) for n = 1..12975
Carlos Rivera, Problem 950: Bi-truncatable primes, The Prime Puzzles & Problems Connection.
Seth A. Troisi, Selected terms: the first 200 primes, the last 300 primes, and the smallest and largest primes of each length

Cf. A024770 (right-truncatable primes), A024785 (left-truncatable primes), A137812 (left-or-right truncatable primes).

Cf. A077391.

msd={1,2,3,4,5,6,7,8,9}; lsd={1,3,7,9}; Clear[p]; p[1]={2,3,5,7}; p[2]={11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}; p[digits_] := p[digits] = Select[Flatten[Outer[Plus, 10^(digits-1)*msd, 10*p[digits-2], lsd]], PrimeQ]; t={}; k=0; While[Length[t] < 100, k++; t=Join[t, p[k]]]; t (* T. D. Noe, Apr 19 2011 *)
paesQ[n_]:=AllTrue[NestWhileList[FromDigits[Most[Rest[ IntegerDigits[ #]]]]&, n,#>99&],PrimeQ]; Select[Prime[Range[150]],paesQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 01 2015 *)

from itertools import count, islice
from sympy import isprime, primerange
def agen(): # generator of terms
    odds, evens, digits = [2, 3, 5, 7], list(primerange(10, 100)), 3
    yield from odds + evens
    while len(odds) > 0 or len(evens) > 0:
        new = []
        old = odds if digits%2 == 1 else evens
        for first in "123456789":
            for p in old:
                mid = str(p)
                for last in "1379":
                    t = int(first + mid + last)
                    if isprime(t):
                        yield t
                        new.append(t)
        old = new
        if digits%2: odds = old
        else: evens = old
        print("...", digits, time()-time0)
        digits += 1
print(list(islice(agen(), 80))) # Michael S. Branicky, May 06 2022

Corrected and extended by T. D. Noe, Apr 19 2011

cp's OEIS Frontend

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

A227916 Primes that remain prime when the leftmost digit is removed.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A024781 Every suffix prime and no 0 digits in base 6 (written in base 6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Python

A024783 Every suffix prime and no 0 digits in base 8 (written in base 8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Python

A024784 Every suffix prime and no 0 digits in base 9 (written in base 9).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Python

A024779 Every suffix is prime and contains no 0 digits in base 4 (written in base 4).

Views

Author

Keywords

Crossrefs

Programs

Python

A024780 Every suffix prime and no 0 digits in base 5 (written in base 5).

Views

Author

Keywords

Crossrefs

Programs

Python

A024782 Every suffix prime and no 0 digits in base 7 (written in base 7).

Views

Author

Keywords

Crossrefs