A024770 -id:A024770 - cp's OEIS frontend

A238854 Largest right-truncatable, reversible prime in base n.

23, 53, 449, 191, 1171, 30671, 5827, 3733, 901687, 10357, 834469, 3043427, 5430889, 4060019, 498061, 34763, 118248433, 62344463, 218555173, 4463351, 114607657, 7903613, 14523874693, 211675817, 32814697, 93375223, 162466979, 8052409793, 12006877873

Offset: 3

Stanislav Sykora, Mar 07 2014

See A238850, A238851, A238852, A238853 for the finite lists of such numbers in four bases selected as examples. A sequence conceptually similar to this one, but for right-truncatable (not reversible!) primes is A023107. The present, more restrictive, condition leads to smaller numbers which can be evaluated in reasonable time for much higher n values.

			a(4) = 53 because it is a prime which in base 4 reads 311_b4, its reverse 113_b4 (decimal 23) is also a prime, the same holds for all its base-4 prefixes (31_b4 and 3_b4), and it is the largest natural having these properties.

Stanislav Sykora, Table of n, a(n) for n = 3..390
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
Cf. In base n: A238855 (totals), A238856 (maximum digits), A238857 (m-digits counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
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A238855 Number of all right-truncatable reversible primes in base n.

Original entry on oeis.org

0, 3, 4, 12, 5, 12, 24, 17, 16, 33, 22, 29, 50, 39, 40, 39, 24, 65, 80, 100, 58, 58, 69, 122, 101, 90, 83, 125, 114, 133, 114, 122, 255, 203, 252, 123, 152, 221, 202, 308, 131, 250, 299, 397, 303, 143, 201, 484, 497, 423, 269, 253, 442, 944, 845, 378, 231, 460, 420, 455, 538, 438

Offset: 2

Stanislav Sykora, Mar 07 2014

For definitions and more comments, see A238854 and A238850.

Conjecture: in any base n, the number of right-truncatable reversible primes is finite.

			In bases 10, 16, 100, and 256 (used as examples in the crossrefs) there are, respectively, 16, 40, 1552, and 35127 such numbers.

Stanislav Sykora, Table of n, a(n) for n = 2..390
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments

Cf. Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
Cf. In base n: A238854 (largest), A238856 (maximum digits), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
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```

A238856 Number of digits of the largest right-truncatable reversible prime in base n.

Original entry on oeis.org

0, 3, 3, 4, 3, 4, 5, 4, 4, 6, 4, 6, 6, 6, 6, 5, 4, 7, 6, 7, 5, 6, 5, 8, 6, 6, 6, 6, 7, 7, 6, 6, 8, 6, 8, 7, 8, 8, 7, 8, 7, 8, 8, 8, 8, 7, 7, 9, 9, 8, 7, 8, 10, 10, 9, 8, 6, 9, 8, 7, 9, 9, 9, 9, 11, 8, 7, 9, 10, 9, 10, 9, 9, 11, 10, 10, 9, 9, 8, 9, 9, 8, 10, 10, 10, 9, 9, 9, 10, 11

Offset: 2

Stanislav Sykora, Mar 13 2014

For definitions and more comments, see A238854 and A238850. A weak conjecture: this sequence might be bounded.

			a(16) = 6 because the largest truncatable reversible prime in base 16 has 6 hexadecimal digits (see A238851).

Stanislav Sykora, Table of n, a(n) for n = 2..390
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
Cf. In base n: A238854 (largest), A238855 (totals), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
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```

A238857 Array read by rows: row n lists total number of m-digit right-truncatable reversible primes in base n.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 1, 0, 2, 4, 4, 2, 0, 3, 1, 1, 0, 3, 5, 3, 1, 0, 4, 7, 7, 5, 1, 0, 4, 5, 5, 3, 0, 4, 5, 6, 1, 0, 4, 8, 7, 9, 4, 1, 0, 5, 5, 7, 5, 0, 5, 10, 8, 4, 1, 1, 0, 6, 11, 17, 12, 3, 1, 0, 6, 11, 13, 6, 2, 1, 0, 6, 9, 11, 9, 4, 1, 0, 6, 13, 12, 7, 1, 0, 7, 9, 7, 1, 0

Offset: 2

Stanislav Sykora, Mar 13 2014

For definitions and more comments, see A238854 and A238850.

This is an irregular table with one line for every base, starting at 2, while the columns correspond to the number of digits (1,2,3,...). Each row terminates with a zero (in any given base there appears to be a finite number of instances).

			These are the first rows of the table:
   2: 0,
   3: 1, 1, 1, 0,
   4: 2, 1, 1, 0,
   5: 2, 4, 4, 2, 0,
   6: 3, 1, 1, 0,
   7: 3, 5, 3, 1, 0,
   8: 4, 7, 7, 5, 1, 0,
   9: 4, 5, 5, 3, 0,
  10: 4, 5, 6, 1, 0,
  ...
Hence, there are 6 right truncatable reversible primes with 3 digits in base 10 (see A238850).

Stanislav Sykora, Table of n, a(n) for n = 2..5071
Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Full in base 10: A238850, 16: A238851, 100: A238852, 256: A238853.
In base n: A238854 (largest), A238855 (totals), A238856 (maximum digits).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
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```

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

A202260 Right-truncatable composites: every decimal prefix is a composite number.

Original entry on oeis.org

4, 6, 8, 9, 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, 400, 402, 403, 404, 405, 406, 407, 408, 420, 422, 423, 424, 425, 426, 427, 428, 429, 440, 441, 442, 444, 445, 446, 447, 448

Offset: 1

Jaroslav Krizek, Dec 25 2011

Subsequence of A202259.

Stanislav Sykora, Table of n, a(n) for n = 1..10000

Cf. A012883 (right-truncatable noncomposites), A202259 (right-truncatable nonprimes), A024770 (right-truncatable primes).

Cf. A254750, A254752, A254754, A254755 (left-truncatable composites).

isComposite(n) = (n>2)&&(!isprime(n));
isRightTruncatableComposite(n,b=10) = {my(k=b);if(!isComposite(n),return(0););while(n\k>0,if(!isComposite(n\k),return(0););k*=b);return(1);} \\ Stanislav Sykora, Feb 15 2015

A238850 Right-truncatable reversible primes in base 10.

Original entry on oeis.org

2, 3, 5, 7, 31, 37, 71, 73, 79, 311, 313, 373, 733, 739, 797, 3733

Offset: 1

Stanislav Sykora, Mar 06 2014

In a general base b, a number qualifies as a member iff: (i) it is a prime, (ii) when its digits in base b are reversed, it is still a prime, and (iii) when, in base b, it has more than one digit and the least significant one is dropped, the remaining prefix has the same properties. This implies that any base-b prefix of such a number, no matter how many right-side digits are truncated, is still a right-truncatable reversible prime. Sequences of this type appear to be all finite (see A238854, A238855, and A238856, used as examples).
This particular sequence is for base b = 10.
See also A238854 for comments on a more general context.

			739 is a member because it is a prime and so is 937, as well as the pair (73, 37) and 7.

Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

In base 16: A238851, 100: A238852, 256: A238853.
In base n: A238854 (largest), A238855 (totals), A238856 (maximum digits), A238857 (m-digit counts).
Cf. A007500, A023107, A024770, A237600, A237601, A237602.

PARI
```
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```

A238851 Right-truncatable, reversible primes in base 16.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 53, 59, 61, 83, 89, 113, 179, 191, 211, 863, 947, 977, 983, 991, 1429, 1439, 1823, 3061, 3067, 3389, 15161, 15643, 15733, 15737, 15739, 15859, 23029, 48989, 48991, 251737, 251831, 253751, 368471, 4060019

Offset: 1

Stanislav Sykora, Mar 06 2014

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

These numbers are fully right-truncatable and reversible primes in base 16 (but listed in decimal format). They are 40 in all.

			The largest such number (4060019) is in hex format 0x3DF373. It is a prime, so is 0x373FD3, and 0x3DF37 has again the same properties.

Stanislav Sykora, PARI/GP scripts for genetic threads, with code and comments.

Cf. All in base 10: A238850, 100: A238852, 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
```
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```

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

Original entry on oeis.org

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);

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A238854 Largest right-truncatable, reversible prime in base n.

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A238855 Number of all right-truncatable reversible primes in base n.

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A238856 Number of digits of the largest right-truncatable reversible prime in base n.

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A238857 Array read by rows: row n lists total number of m-digit right-truncatable reversible primes in base n.

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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.

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A202260 Right-truncatable composites: every decimal prefix is a composite number.

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A238850 Right-truncatable reversible primes in base 10.

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A238851 Right-truncatable, reversible primes in base 16.

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