A024785 -id:A024785 - cp's OEIS frontend

A132394 Left-truncatable primes in the order generated (with no zero digits).

2, 3, 5, 7, 13, 23, 43, 53, 73, 83, 17, 37, 47, 67, 97, 113, 313, 613, 223, 523, 823, 443, 643, 743, 353, 653, 853, 953, 173, 373, 673, 773, 283, 383, 683, 883, 983, 317, 617, 137, 337, 937, 347, 547, 647, 947, 167, 367, 467, 967, 197, 397, 797, 997, 2113

Offset: 1

Harry J. Smith, Nov 11 2007

The most logical way of generating the Left-truncatable primes generates them in this order. Last term is a(4260)=357686312646216567629137.

H. J. Smith, Table of n, a(n) for n = 1..4260 (full sequence)
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
H. J. Smith, Link to program to generate this sequence and its output file.
Index entries for sequences related to truncatable primes

Cf. A024785.

{fileO="b132394.txt";v=vector(4260);v[1]=2;v[2]=3;v[3]=5;v[4]=7;i=0;j=4; write(fileO,"1 2");write(fileO,"2 3");write(fileO,"3 5");write(fileO,"4 7"); until(i>=j,i++;p=v[i];P10=10^(1+log(p)\log(10)); for(k=1,9,z=k*P10+p;if(isprime(z),j++;v[j]=z;write(fileO,j," ",z);)));} \\ Harry J. Smith, Sep 18 2008

Edited by Jason G. Wurtzel, Oct 06 2010

A383781 Primes where successively deleting the most significant digit yields a sequence that alternates between a prime and a nonprime at every step until a single-digit number remains.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 127, 157, 163, 193, 227, 233, 257, 263, 277, 293, 433, 457, 463, 487, 557, 563, 577, 587, 593, 677, 727, 733, 757, 787, 827, 857, 863, 877, 887, 977, 1129, 1171, 1231, 1259, 1279, 1289, 1319, 1361, 1429, 1459

Offset: 1

Stefano Spezia, May 09 2025

Is this sequence infinite?

See A383782 and the comment therein. - Michael S. Branicky, May 11 2025

			127 is a term since 127 is a prime, 27 is a nonprime, and 7 is a prime;
13 is not a term since 13 and 3 are both prime.

Cf. A024785, A383780, A383782.

Unprotect[CompositeQ]; CompositeQ[1]:=True; Protect[CompositeQ]; Q[n_]:=And[AllTrue[FromDigits/@Table[Take[IntegerDigits[n], -i], {i,IntegerLength[n],1,-2}], PrimeQ], AllTrue[FromDigits/@Table[Take[IntegerDigits[n], -i], {i,IntegerLength[n]-1,1,-2}], CompositeQ]]; Select[Prime[Range[240]], Q]

from gmpy2 import is_prime, mpz
from itertools import count, islice
def agen():
    olst, elst = [2, 3, 5, 7], [11, 19, 29, 31, 41, 59, 61, 71, 79, 89]
    for n in count(1):
        yield from sorted(olst + elst)
        olst2, elst2 = [], []
        for o in olst:
            o, base = o, 10**(2*n-1)
            for i in range(10*base, 100*base, base):
                t = i + o
                t2 = int(str(t)[1:])
                if is_prime(t) and not is_prime(t2):
                    olst2.append(t)
        for e in elst:
            e, base = e, 10**(2*n)
            for i in range(10*base, 100*base, base):
                t = i + e
                t2 = int(str(t)[1:])
                if is_prime(t) and not is_prime(t2):
                    elst2.append(t)
        olst, elst = olst2, elst2
print(list(islice(agen(), 68))) # Michael S. Branicky, May 11 2025

A032449 Every suffix is prime and no digit is 0 or 5.

Original entry on oeis.org

2, 3, 7, 13, 17, 23, 37, 43, 47, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 367, 373, 383, 397, 443, 467, 613, 617, 643, 647, 673, 683, 743, 773, 797, 823, 883, 937, 947, 967, 983, 997, 1223, 1283, 1367, 1373, 1613, 1823, 1997

Offset: 1

Carlos Rivera

The last term in the sequence is a(2003) = 666276812967623946997.

Nathaniel Johnston, Table of n, a(n) for n = 1..2003 (full sequence)

Cf. A024785.

a:=[[2],[3],[7]]: s:=[1,2,3,4,6,7,8,9]: l1:=1: l2:=3: for n from 1 to 4 do for k from 1 to 8 do for j from l1 to l2 do d:=[op(a[j]),s[k]]: 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

Corrected terms and statement about last term, Nathaniel Johnston, Jun 22 2011.

A144714 Left-truncatable primes that contain one or more zero digits.

Original entry on oeis.org

103, 107, 307, 503, 607, 907, 1013, 1097, 1103, 1307, 1607, 1907, 2003, 2017, 2053, 2083, 2503, 3023, 3037, 3067, 3083, 3307, 3607, 3907, 4003, 4007, 4013, 4073, 5003, 5023, 5107, 5503, 6007, 6037, 6043, 6047, 6053, 6067, 6073, 6607, 6907, 7013, 7043

Offset: 1

Harry J. Smith, Oct 08 2008

These are the terms in sequence A033664 that are not in A024785. This sequence is infinitely long.

H. J. Smith, 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.
H. J. Smith, Link to programs to generate truncatable primes.
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Cf. A020994, A024770, A024785, A033664, A132394.

zeroin(z)={until(z==0,q=z\10;r=z-10*q;if(r==0,return(1));z=q;);return(0);}
{fileO="b144714.txt";v=vector(15000);v[1]=2;v[2]=3;v[3]=5;v[4]=7;j=4;m=0;
p10=1;until(0,p10*=10;j0=j;for(k=1,9,k10=k*p10;for(i=1,j0,z=k10+v[i];
if(isprime(z),j++;v[j]=z;if(zeroin(z),m++;
write(fileO,m," ",z);if(m==10000,break(3));)))));}

A173057 Partial sums of A024770.

Original entry on oeis.org

2, 5, 10, 17, 40, 69, 100, 137, 190, 249, 320, 393, 472, 705, 944, 1237, 1548, 1861, 2178, 2551, 2930, 3523, 4122, 4841, 5574, 6313, 7110, 9443, 11782, 14175, 16574, 19513, 22632, 25769, 29502, 33241, 37034, 40831, 46770, 53963, 61294, 68627

Offset: 1

Jonathan Vos Post, Feb 08 2010

Partial sums of right-truncatable primes, primes whose every prefix is prime (in decimal representation). The sequence has 83 terms. The subsequence of prime partial sums of right-truncatable primes begins: 2, 5, 17, 137, 1237, 1861, 2551, 199483. What is the largest value in the subsubsequence of right-truncatable prime partial sums of right-truncatable primes?

			a(50) = 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 + 31379.

Cf. A000040, A024770, A033664, A024785 (left-truncatable primes), A032437, A020994, A052023, A052024, A052025, A050986, A050987, A069866.

A240768 Left-truncatable primes p with property that prepending any single decimal digit to p does not produce a prime.

Original entry on oeis.org

2, 5, 773, 3373, 3947, 4643, 5113, 6397, 6967, 7937, 15647, 16823, 24373, 33547, 34337, 37643, 56983, 57853, 59743, 62383, 63347, 63617, 69337, 72467, 72617, 75653, 76367, 87643, 92683, 97883, 98317, 121997, 124337, 163853, 213613, 236653, 242467, 242797

Offset: 1

Arkadiusz Wesolowski, Apr 12 2014

			3373 belongs to this sequence because 3373, 373, 73 and 3 are all prime; k*10^4 + 3373, for k = 1 to 9, are all composite.

Chris Caldwell, The Prime Glossary, Left-truncatable prime
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Subsequence of A024785 and of A155762.

for(n=2, 242797, v=n; while(isprime(n), c=n; n=lift(Mod(c, 10^(#Str(c)-1))); if(!(#Str(c)-#Str(n)==1), break)); if(n==0, s=#Str(v); t=0; for(k=1, 9, if(isprime(k*10^s+v), break, t++)); if(t==9, print1(v, ", "))); n=v);

A024785 INTERSECT A155762.

A253386 Suffixes of 3991687693967 (left-truncatable prime).

Original entry on oeis.org

7, 67, 967, 3967, 93967, 693967, 7693967, 87693967, 687693967, 1687693967, 91687693967, 991687693967, 3991687693967

Offset: 1

Mikk Heidemaa, Dec 31 2014

3991687693967 (13 digits) includes the longest (7 digits) palindromic prime suffix (7693967) among the left-truncatable primes (digit '0' excluded). The largest one (24 digits, see A253427) contains a nonprime palindrome of 7 digits (1264621). The terms from a(3) to a(13) cannot be written as a sum of 3 squares.

			Triangular form:
----------------
............7
...........67
..........967*
.........3967
........93967
.......693967
......7693967**
.....87693967
....687693967
...1687693967
..91687693967
.991687693967
3991687693967***
----------------
* None from 3rd row (967,...,3991687693967) cannot be written as a sum of 3 squares.
** The palindromic prime suffix.
*** a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g * h * i = 3991687693967;
a=693967; b=93967; c=3967; d=967; e=67; f=7; g=4114278523; h=37; i=27.
a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g * h * i = 3991687693967^5;
a=693967; b=93967; c=3967; d=967; e=67; f=7;
g=3571123727278334614405609468109056139549629; h=228288322626423; i=124339.
(All primes.)

Mikk Heidemaa, Related material (2015)
Eric Weisstein's World of Mathematics, Truncatable Prime
Wikipedia, Truncatable prime
Index entries for sequences related to truncatable primes

Cf. A012885, A024785, A253427.

Column[ Table[ Mod[ 3991687693967, 10^n], {n, 13}], Right] (* Mikk Heidemaa, Oct 07 2017 *)

a(n) = 3991687693967 mod 10^n for 1 <= n <= 13. - Mikk Heidemaa, Oct 07 2017

A253427 Number of left-truncatable n-digit primes with least significant digit 7.

Original entry on oeis.org

1, 5, 17, 47, 89, 150, 199, 238, 264, 257, 230, 181, 139, 117, 70, 46, 33, 19, 11, 5, 5, 4, 3, 1

Offset: 1

Mikk Heidemaa, Dec 31 2014

Number of n-digit (n <= 24) truncatable primes (i.e., left-truncatable with the least significant digit 7; only prime suffixes without digit '0'). Formed by successively prepending a digit (1,...,9) to the most significant digit (1st is 7) while preserving primality; a(13) with 139 members includes A253386 (includes the palindromic prime suffix 7693967), but a(24) includes just a single one, 357686312646216567629137 (contains the nonprime palindrome 1264621).

			a(1)=1 {7};
a(2)=5 {17,37,47,67,97};
...
a(23)=3 {57686312646216567629137, 95918918997653319693967, 96686312646216567629137};
a(24)=1 {357686312646216567629137}; see also A012885.

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

Cf. A012885, A024785, A050987, A253386.

S[1]:= {7}:
for n from 2 while nops(S[n-1]) > 0 do
  S[n]:= select(isprime, {seq(seq(i*10^(n-1)+s, i=1..9),s = S[n-1])})
od:
seq(nops(S[i]),i=1..n-2); # Robert Israel, Jan 01 2015

A308711 Left-truncatable primes in base-10 bijective numeration.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 103, 107, 113, 137, 167, 173, 197, 223, 283, 307, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 503, 523, 547, 607, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 907, 937, 947, 953, 967, 983, 997

Offset: 1

Robin Houston, Jun 19 2019

Not identical to A033664; in fact a strict subsequence of A033664. For example, 2003 belongs to A033664 but not to this sequence, since in bijective numerals 2003 is 19X3, whose suffix 9X3 = 1003 = 17 * 59.

Robin Houston, Table of n, a(n) for n = 1..8391
Wikipedia, Bijective numeration

Cf. A024785, A033664.

DIGITS = "123456789X"
DECODE = {d: i + 1 for i, d in enumerate(DIGITS)}
def decode(s):
    return reduce(lambda n, c: 10 * n + DECODE[c], s, 0)
def search(s):
    n = decode(s)
    if n > 0:
        if not is_prime(n): return
        yield n
    for digit in DIGITS: yield from search(digit + s)
full = sorted(search(""))
full[:10]

A385404 Numbers that can be split into two at any place between their digits such that the resulting numbers are always a nonprime on the left and a prime on the right.

Original entry on oeis.org

12, 13, 15, 17, 42, 43, 45, 47, 62, 63, 65, 67, 82, 83, 85, 87, 92, 93, 95, 97, 123, 143, 147, 153, 167, 183, 423, 443, 447, 453, 467, 483, 497, 623, 637, 643, 647, 653, 667, 683, 697, 813, 817, 823, 843, 847, 853, 867, 873, 883, 913, 917, 923, 937, 943, 947, 953, 967, 983, 997

Offset: 1

Tamas Sandor Nagy, Jun 27 2025

As no leading zeros are allowed, all terms are zeroless.
From Michael S. Branicky, Jun 27 2025: (Start)
Finite since each term with first digit removed must be a member of A024785, which is finite.
Last term here is a(7407) = 6357686312646216567629137. (End)

			637 is a term because when it is split in two in all possible ways, it first results in 63, a nonprime, and 3, a prime. When split in the second and final possible way, it results in 6, a nonprime, and 37, a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..7407

Cf. A125664, A125524, A024785.

q[n_] := !MemberQ[IntegerDigits[n], 0] && AllTrue[Range[IntegerLength[n]-1], PrimeQ[QuotientRemainder[n, 10^#]] == {False, True} &]; Select[Range[10, 1000], q] (* Amiram Eldar, Jun 27 2025 *)

from sympy import isprime
def ok(n): return '0' not in (s:=str(n)) and len(s) > 1 and all(not isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, Jun 27 2025

# uses import and function ok above
from itertools import count, islice, product
def agen():  # generator of terms
    tp = list("23579")  # set of left-truncatable primes
    for d in count(2):
        tpnew = []
        for f in "123456789":
            for e in tp:
                if isprime(int(s:=f+e)):
                    tpnew.append(s)
                if ok(t:=int(f+e)):
                    yield t
        tp = tpnew
        if len(tp) == 0:
            return
afull = list(agen())
print(afull[:60]) # Michael S. Branicky, Jun 27 2025

cp's OEIS Frontend

A132394 Left-truncatable primes in the order generated (with no zero digits).

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A383781 Primes where successively deleting the most significant digit yields a sequence that alternates between a prime and a nonprime at every step until a single-digit number remains.

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A032449 Every suffix is prime and no digit is 0 or 5.

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A144714 Left-truncatable primes that contain one or more zero digits.

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PARI

A173057 Partial sums of A024770.

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A240768 Left-truncatable primes p with property that prepending any single decimal digit to p does not produce a prime.

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PARI

Formula

A253386 Suffixes of 3991687693967 (left-truncatable prime).

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Mathematica

Formula

A253427 Number of left-truncatable n-digit primes with least significant digit 7.

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Programs

Maple

A308711 Left-truncatable primes in base-10 bijective numeration.

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