A024785 -id:A024785 - cp's OEIS frontend

A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 237, 297, 313, 317, 373, 537, 597, 713, 717, 737, 797, 2337, 2397, 2937, 3113, 3137, 3173, 3797, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313, 739397

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of a(n) in any way into two nonempty parts, each part represents a prime number.
Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since these are finite sequences, a(n) is also finite. It has 45 members, the largest of which is 739397 and happens to be a prime.
The sequence is a union of A254753 and A020994.
A subsequence of A260181. - M. F. Hasler, Sep 16 2016

			6 is not a member because its expansion cannot be sliced in two.
597 is a member because (5,97,59, and 7) are all primes.
2331 is excluded because 233 is prime, but 1 is not. - _Gordon Hamilton_, Feb 20 2015

Cf. A020994, A024770, A024785, A254750, A254752, A254753, A254754, A254756.

Cf. A260181.

fQ[n_] := (p = {2, 3, 5, 7}; If[ Union@ Join[p, {Mod[n, 10]}] != p, {False}, Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, Union@ PrimeQ@ Flatten@ Table[{FromDigits[ Take[idn, i]], FromDigits[ Take[idn, -lng + i - 1]]}, {i, lng}] == {True}]]); Select[ Range@1000000, fQ] (* Robert G. Wilson v, Feb 21 2015 *)
Select[Range[10,750000],AllTrue[Flatten[Table[FromDigits/@TakeDrop[IntegerDigits[#],n],{n,IntegerLength[#]-1}]],PrimeQ]&] (* Harvey P. Dale, Feb 13 2024 *)

slicesIntoPrimes(n,b=10) = {my(k=b);if(n0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);return(1);}

def breakIntoPrimes(n):
    D=n.digits()
    for i in [1..len(D)-1]:
        if not(is_prime(sum(D[i:][j]*10^j for j in range(len(D[i:])))) and is_prime(sum(D[:i][j]*10^j for j in range(len(D[:i]))))):
            return False
        else:
            continue
    return True
[n for n in [10..1000] if breakIntoPrimes(n)] # Tom Edgar, Feb 20 2015

A254753 Composite numbers with only prime proper prefixes and suffixes in base 10.

Original entry on oeis.org

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 297, 537, 597, 713, 717, 737, 2337, 2397, 2937, 3113, 3173, 5937, 5997, 7197, 7337, 7397, 29397, 31373, 37937, 59397, 73313

Offset: 1

Stanislav Sykora, Feb 15 2015

A proper prefix (or suffix) of a number m is one which is neither void, nor identical to m.
Alternative definition: Slicing the decimal expansion of a composite a(n) in any way into two nonempty parts, each part represents a prime number.
This sequence is a subset of A254751. Every proper prefix of each member a(n) is a member of A024770, and every proper suffix is a member of A024785. Since the latter are finite sequences, a(n) is also a finite sequence. It has 34 members, the largest of which is the composite number 73313.
Should one change the definition to 'prime numbers such that, in base 10, all their proper prefixes and suffixes represent primes', the result would be the sequence A020994.

			6 is not a member because its expansion cannot be sliced in two.
The composite 73313 is a member because (7, 3313, 73, 313, 733, 13, 7331, 3) are all primes.

Cf. A020994, A024770, A024785, A254750, A254751, A254752, A254754.

apQ[n_]:=Module[{idn=IntegerDigits[n],c1,c2},c1=FromDigits/@ Table[ Take[ idn,k],{k,Length[idn]-1}];c2=FromDigits/@Table[Take[idn,k],{k,-(Length[ idn]-1), -1}]; AllTrue[ Join[c1,c2],PrimeQ]]; Select[Range[ 10,80000], CompositeQ[#] && apQ[#]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 29 2018 *)

isComposite(n) = (n>2)&&(!isprime(n));
slicesIntoPrimes(n,b=10) = {my(k=b);if(n0,if(!isprime(n\k)||!isprime(n%k),return(0););k*=b;);1;}
isCompositeSlicingIntoPrimes(n,b=10) = isComposite(n) && slicesIntoPrimes(n,b);

A101115 Beginning with the n-th prime, the number of successive times a new prime can be formed by prepending the smallest nonzero digit.

Original entry on oeis.org

0, 5, 0, 9, 5, 4, 8, 4, 5, 9, 4, 6, 2, 7, 6, 8, 9, 7, 6, 3, 14, 5, 5, 2, 4, 10, 1, 5, 7, 3, 4, 3, 5, 5, 0, 6, 5, 8, 5, 13, 4, 5, 4, 5, 3, 8, 4, 4, 5, 8, 3, 6, 1, 4, 4, 2, 5, 2, 2, 3, 4, 9, 8, 7, 4, 7, 3, 3, 5, 5, 7, 8, 4, 3, 3, 2, 1, 7, 0, 4, 3, 5, 3, 7, 9, 6, 6, 5, 6, 8

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Dec 02 2004

It is possible the procedure described would generate some left-truncatable primes (A024785). Although zero digits cannot be added, it is possible the starting prime may contain zeros. Therefore the possible number of digit additions is not limited by the length of the largest known left-truncatable prime. Further, because the smallest digit that satisfies the requirement is used each time, it is possible that choosing a larger digit would allow more single digits to be added. Therefore although some of the set of left-truncatable primes may be generated by this practice, not all of them will.

In principle it is possible that some a(n) is undefined because the process could go on indefinitely, but this is very unlikely. The largest a(n) for n <= 300000 is a(49120) = 18. - Robert Israel, Jun 29 2015

			a(2) is 5 because the second prime is 3, to which single nonzero digits can be prepended 5 times yielding a new prime each time (giving preference to the smallest digit that satisfies the requirement): 13, 113, 2113, 12113, 612113 (see A053583). There is no nonzero digit which can be prepended to 612113 to yield a new prime.
a(21) = 14 because the 21st prime (73) can be prepended with single nonzero digits 14 times yielding a new prime each time: 73, 173, 6173, 66173, ..., 4818372912366173.

Robert Israel, 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.
Index entries for sequences related to truncatable primes

Cf. A053583, A024785, A000040, A101116, A101117, A101118.

f:= proc(n) local p, nd, d, count, x, success;
      p:= ithprime(n); nd:= ilog10(p);
      for count from 0 do
        nd:= nd+1;
        success:= false;
        for d from 1 to 9 do
          x:= 10^nd * d + p;
          if isprime(x) then
             success:= true;
             break
          fi
        od;
        if not success then return(count) fi;
        p:= x;
      od
end proc:
map(f, [$1..100]); # Robert Israel, Jun 29 2015

from sympy import isprime, prime
def a(n):
    pn = prime(n)
    s, c, found = str(pn), 0, True
    while found:
        found = False
        for d in "123456789":
            if isprime(int(d+s)):
                s, c, found = d+s, c+1, True
                break
    return c
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Jun 24 2022

A101116 Values in A101115 which are records.

Original entry on oeis.org

0, 5, 9, 14, 15, 18, 19, 20, 22

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Dec 02 2004

All primes up to 1000003 have been tested.

Cf. A053583, A024785, A000040, A101115, A101117, A101118.

from sympy import isprime, nextprime
def agen(): # generator of tuple of terms of (A101116, A101117, A101118)
    n, pn, record = 0, 1, -1
    while True:
        n += 1
        pn = nextprime(pn)
        s, c, found = str(pn), 0, True
        while found:
            found = False
            for d in "123456789":
                if isprime(int(d+s)):
                    s, c, found = d+s, c+1, True
                    break
        if c > record:
            record = c
            yield record, pn, int(s)
g = agen()
print([next(g)[0] for n in range(1, 7)]) # Michael S. Branicky, Jun 24 2022

a(7)-a(8) from Michael S. Branicky, Jun 24 2022

a(9) from Michael S. Branicky, Jul 26 2024

A101118 a(n) = the resulting prime generated when the process described in A101115 is applied to A101117(n).

Original entry on oeis.org

2, 612113, 5372126317, 4818372912366173, 21291981879276213799, 912733515196363393600307, 668334992181698187977197951, 231879245133561335194866134641, 933651219687395363156136052921903

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Dec 02 2004

			a(3) = 5372126317 because 5372126317 is the last prime that can be generated by successively prepending nonzero digits to A101117(3). A101117(3) is 7. A101116(3) indicates that 9 digits can be successively prepended to 7 generating a new prime each time. Doing so and giving preference to the smallest digit which meets the requirement, generates the following primes: 17, 317, 6317, 26317, 126317, 2126317, 72126317, 372126317, 5372126317.

Cf. A053583, A024785, A000040, A101115, A101116, A101117.

g = agen() # uses agen() and imports from A101116
print([next(g)[2] for n in range(1, 7)]) # Michael S. Branicky, Jun 24 2022

a(7)-a(8) and typo corrected in a(6) from Michael S. Branicky, Jun 24 2022

a(9) from Michael S. Branicky, Jul 26 2024

A012885 Suffixes of 357686312646216567629137 (all primes).

Original entry on oeis.org

7, 37, 137, 9137, 29137, 629137, 7629137, 67629137, 567629137, 6567629137, 16567629137, 216567629137, 6216567629137, 46216567629137, 646216567629137, 2646216567629137, 12646216567629137, 312646216567629137, 6312646216567629137, 86312646216567629137

Offset: 1

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

From Mikk Heidemaa, Mar 23 2015: (Start)
a(2),...,a(24) all have a single representation (in positive integers) as the sum of two squares (e.g., a(24) = 416865370156^2 + 428846797599^2) and as the hypotenuse of a primitive Pythagorean triple (357686312646216567629137^2 = 10132838975618776700465^2 + 357542758042644694110888^2).
---
a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 357686312646216567629137^2;
a=304233682432674451033719; b=185074861663432734470527;
c=4189176178164916432878; d=33333333333333333333333; e=3333333; f=3.
---
a^2 + b^2 + c^2 + d^2 + e^2 + f^2 = 357686312646216567629137^3;
a=210197737649788368191109924028342434;
b=39738123500625252940689952285037741;
c=777777777777777777777777;
d=777777777777777777777777;
e=777777777777777777;
f=777777777777.
a=170350493188466620042802284807886346;
b=129394423538599186274382140531063939;
c=777777777777777777777777;
d=777777777777777777777777;
e=777777777777777777;
f=777777777777.
---
x^2 + y^2 = 357686312646216567629137^3;
x=144701758632763782416276428525674993;
y=157555096461604743754426503960480452;
x=149107037120999813337660002835835372;
y=153392629723324670471173010334042063.
(End)

			.......................7
......................37
.....................137
....................9137
...................29137
..................629137
.................7629137
................67629137
...............567629137
..............6567629137
.............16567629137
............216567629137
...........6216567629137
..........46216567629137
.........646216567629137
........2646216567629137
.......12646216567629137
......312646216567629137
.....6312646216567629137
....86312646216567629137
...686312646216567629137
..7686312646216567629137
.57686312646216567629137
357686312646216567629137
------------------------

Mikk Heidemaa, Table of n, a(n) for n = 1..24
James Grime and Brady Haran, 357686312646216567629137, Numberphile video (2018).
Miguel A. Martin-Delgado, Chiral Prime Concatenations, arXiv:2009.12305 [math.NT], 2020.

Cf. A024785.

Table[Mod[357686312646216567629137,10^n] ,{n, 1, 24}] (* José de Jesús Camacho Medina, Dec 21 2016 *)

a(n) = 357686312646216567629137 mod 10^n. - José de Jesús Camacho Medina, Dec 21 2016

A055521 Restricted left truncatable (Henry VIII) primes.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

There are 1440 such primes, the largest being 357686312646216567629137.

Left-truncatable primes (A024785) which have at least two digits and are not the end of a larger left-truncatable prime. - Jens Kruse Andersen, Jul 29 2014

			773 is in the sequence since 773, 73, 3 are primes, while no digit 1..9 gives a prime if placed before 773. 13 is not in the sequence since for example 113 is prime. 2 and 5 are disqualified for only having one digit. - _Jens Kruse Andersen_, Jul 29 2014

Kahan, S. and Weintraub, S. "Left Truncatable Primes." J. Recr. Math. 29, 254-264, 1998.

Jens Kruse Andersen, Table of n, a(n) for n = 1..1440 (complete sequence)
I. O. Angell, and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
James Grime and Brady Haran, 357686312646216567629137, Numberphile video (2018)
Eric Weisstein's World of Mathematics, Truncatable Prime
Index entries for sequences related to truncatable primes

Cf. A024785.

from sympy import isprime, primerange
def afull():
    alst, prime_strs, an, digits = [], ["2", "3", "5", "7"], 0, 1
    while len(prime_strs) > 0:
        new_prime_strs = set()
        for p in prime_strs:
            can_extend = False
            for d in "123456789":
                c = d + p
                if isprime(int(c)):
                    can_extend = True
                    new_prime_strs.add(c)
            if digits > 1 and not can_extend:
                alst.append(int(p))
        prime_strs = new_prime_strs
        digits += 1
    return sorted(alst)
print(afull()) # Michael S. Branicky, Dec 11 2022

A101117 a(n) = the first prime yielding the record value A101116(n).

Original entry on oeis.org

2, 3, 7, 73, 13799, 600307, 77197951, 4866134641, 36052921903

Offset: 1

Chuck Seggelin (seqfan(AT)plastereddragon.com), Dec 02 2004

			a(6) = 600307 because 600307 is the first prime to which A101116(6) digits (18) can be prepended yielding a new prime each time (giving preference to the smallest digit which meets the requirement) - 600307, 3600307, 93600307, ..., 9912733515196363393600307.

Cf. A053583, A024785, A000040, A101115, A101116, A101118.

g = agen() # uses agen() and imports from A101116
print([next(g)[1] for n in range(1, 7)]) # Michael S. Branicky, Jun 24 2022

a(7)-a(8) from Michael S. Branicky, Jun 24 2022

a(9) from Michael S. Branicky, Jul 26 2024

A127891 Smallest n-digit left-truncatable prime.

Original entry on oeis.org

2, 13, 113, 1223, 12113, 121283, 1237547, 12184967, 124536947, 1219861613, 12181833347, 121339693967, 1213536676883, 12673876537547, 121848768729173, 1275463876537547, 12429121339693967, 165678739293946997, 1276812967623946997, 15396334245663786197, 315396334245663786197, 5918918997653319693967, 57686312646216567629137, 357686312646216567629137

Offset: 1

Ray Chandler, Feb 04 2007

Ends at a(24) = 357686312646216567629137.

Agrees with A088604 for 24 terms, but this sequence ends there while A088604 continues.

Ray Chandler, Table of n, a(n) for n = 1..24
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Index entries for sequences related to truncatable primes

Cf. A024785, A050987, A088604, A127892.

grow[p_, digits_] := Select[ Table[ FromDigits[Join[{k}, IntegerDigits[p]]], {k, 1, 9}], PrimeQ[#] && Length[ IntegerDigits[#]] == digits&]; g[1] = {3, 7}; g[n_] := g[n] = grow[#, n]& /@ g[n-1] // Flatten; a[1] = 2; a[n_] := Min[g[n]]; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Aug 05 2013 *)

A127892 Largest n-digit left-truncatable prime.

Original entry on oeis.org

7, 97, 997, 9967, 99643, 998443, 9986113, 99979337, 999962683, 9987983617, 99978397283, 999631686353, 9997564326947, 99769267986197, 999769267986197, 9957969395462467, 99957969395462467, 992429121339693967, 8963315421273233617, 86312646216567629137

Offset: 1

Ray Chandler, Feb 04 2007

Ends at a(24) = 357686312646216567629137.

Ray Chandler, Table of n, a(n) for n = 1..24
I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977.
Index entries for sequences related to truncatable primes

Cf. A024785, A050987, A127891.

cp's OEIS Frontend

A254751 Numbers such that, in base 10, all their proper prefixes and suffixes represent primes.

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A254753 Composite numbers with only prime proper prefixes and suffixes in base 10.

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A101115 Beginning with the n-th prime, the number of successive times a new prime can be formed by prepending the smallest nonzero digit.

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A101116 Values in A101115 which are records.

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A101118 a(n) = the resulting prime generated when the process described in A101115 is applied to A101117(n).

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A012885 Suffixes of 357686312646216567629137 (all primes).

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A055521 Restricted left truncatable (Henry VIII) primes.

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A101117 a(n) = the first prime yielding the record value A101116(n).

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