A007500 -id:A007500 - cp's OEIS frontend

A114018 Least n-digit prime whose digit reversal is also prime.

2, 11, 101, 1009, 10007, 100049, 1000033, 10000169, 100000007, 1000000007, 10000000207, 100000000237, 1000000000091, 10000000000313, 100000000000261, 1000000000000273, 10000000000000079, 100000000000000049, 1000000000000002901, 10000000000000000051

Offset: 1

Amarnath Murthy, Nov 12 2005

The more compact version A168159 gives many more terms, cf. formula. [M. F. Hasler, Nov 21 2009]

Michael S. Branicky, Table of n, a(n) for n = 1..500 (terms 1..100 from Harvey P. Dale)

Cf. A168159, A007500, A006567, A122490. [M. F. Hasler, Nov 21 2009]

f[n_] := Block[{k = 10^(n - 1)}, While[ !PrimeQ[k] || !PrimeQ[FromDigits@Reverse@IntegerDigits@k], k++ ]; k]; Array[f, 19] (* Robert G. Wilson v, Nov 19 2005 *)
lndp[n_]:=Module[{p=NextPrime[10^n]},While[CompositeQ[IntegerReverse[ p]],p = NextPrime[ p]];p]; Array[lndp,20,0] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 05 2019 *)

for(x=1,1e99, until( isprime(x=nextprime(x+1)) & isprime(eval(concat(vecextract(Vec(Str(x)),"-1..1")))),); print1(x", "); x=10^#Str(x)-1) \\ M. F. Hasler, Nov 21 2009

from sympy import isprime
def c(n): return isprime(n) and isprime(int(str(n)[::-1]))
def a(n): return next(p for p in range(10**(n-1), 10**n) if c(p))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 27 2022

a(n) = 10^(n-1) + A168159(n). [M. F. Hasler, Nov 21 2009]

More terms from Robert G. Wilson v, Nov 19 2005

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

Original entry on oeis.org

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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See the link.
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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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See the link.
```

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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See the link.
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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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See the link.
```

A359136 Primes such that there is a nontrivial permutation which when applied to the digits produces a prime (Version 1).

Original entry on oeis.org

11, 13, 17, 31, 37, 71, 73, 79, 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, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 419, 421

Offset: 1

N. J. A. Sloane and Allan C. Wechsler, Jan 22 2023

A prime p with decimal expansion p = d_1 d_2 ... d_m is in this sequence iff there is a non-identity permutation pi in S_m such that q = d_pi(1) d_pi(2) ... d_pi(m) is also a prime. The prime q may or may not be equal to p.  Leading zeros are permitted in q.
One must be careful when using the phrase "nontrivial permutation of the digits". When the first and third digits of 101 are exchanged, this is applying the nontrivial permutation (1,3) in S_3 to the digits, leaving the number itself unchanged. One should specify whether it is the permutation that is nontrivial, or its action on what is being permuted. In this sequence and A359137, we mean that the permutation itself is nontrivial.
There are only 34 primes not in this sequence, the greatest of which is 5849. - Andrew Howroyd, Jan 22 2023

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Many OEIS entries are subsequences (possibly after omitting 2, 3, 5, and 7): A007500, A055387, A061461, A069706, A090933, A225035.

Cf. A359137, A359138, A359139.

isok(n)={my(v=vecsort(digits(n))); if(#Set(v)<#v, 1, forperm(v, u, my(t=fromdigits(Vec(u))); if(isprime(t) && t<>n, return(1))); 0)} \\ Andrew Howroyd, Jan 22 2023

from sympy import isprime
from itertools import permutations as P
def ok(n):
    if not isprime(n): return False
    if len(s:=str(n)) > len(set(s)): return True
    return any(isprime(t) for t in (int("".join(p)) for p in P(s)) if t!=n)
print([k for k in range(422) if ok(k)]) # Michael S. Branicky, Jan 23 2023

More than the usual number of terms are shown in order to distinguish this from neighboring sequences.

Incorrect terms removed by Andrew Howroyd, Jan 22 2023

A033294 Squares which when written backwards remain square (final 0's excluded).

Original entry on oeis.org

1, 4, 9, 121, 144, 169, 441, 484, 676, 961, 1089, 9801, 10201, 10404, 10609, 12321, 12544, 12769, 14641, 14884, 40401, 40804, 44521, 44944, 48841, 69696, 90601, 94249, 96721, 698896, 1002001, 1004004, 1006009, 1022121, 1024144, 1026169

Offset: 1

Jeff Burch

Of this sequence's first 10000 terms, only nine have an even number of digits; see A354256.

			144 = 12 * 12 is a term because 441 = 21 * 21.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
Index entry for sequences related to reversing digits of squares

Subsequence of A115690.

Cf. A007488, A007500, A061457, A354256.

a033294 n = a033294_list !! (n-1)
a033294_list = filter chi a000290_list where
  chi m = m `mod` 10 > 0 && head ds `elem` [1,4,5,6,9] &&
          a010052 (foldl (\v d -> 10 * v + d) 0 ds) == 1 where
    ds = unfoldr
         (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 10) m
-- Reinhard Zumkeller, Jan 19 2012

Select[Range[1100]^2,Mod[#,10]!=0&&IntegerQ[Sqrt[FromDigits[Reverse[ IntegerDigits[ #]]]]]&] (* Harvey P. Dale, Oct 28 2013 *)

from math import isqrt
from itertools import count, islice
def sqr(n): return isqrt(n)**2 == n
def agen():
    yield from (k*k for k in count(1) if k%10 and sqr(int(str(k*k)[::-1])))
print(list(islice(agen(), 36))) # Michael S. Branicky, May 21 2022

More terms from Erich Friedman

Initial 0 removed and offset changed by Reinhard Zumkeller, Jan 19 2012

A074833 Primes whose ternary reversal is also prime.

Original entry on oeis.org

2, 5, 7, 11, 13, 19, 23, 31, 37, 41, 47, 53, 61, 67, 79, 83, 101, 103, 113, 127, 131, 151, 163, 167, 173, 179, 181, 191, 193, 211, 227, 233, 263, 281, 293, 311, 331, 349, 353, 359, 401, 409, 419, 421, 431, 439, 449, 463, 467, 491, 499, 503, 521, 523, 541, 563

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074834, A075235, A075236, A075237, A075238, A075239.

Prime[ Select[ Range[110], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 3]], 3]] &]]

A074834 Primes whose base 4 reversal is also prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 23, 29, 31, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 107, 193, 197, 199, 211, 233, 239, 241, 251, 257, 269, 277, 281, 293, 311, 313, 337, 353, 367, 373, 383, 397, 401, 409, 419, 433, 443, 449, 457, 461, 467, 487, 491, 499, 509, 787, 797, 809

Offset: 1

Robert G. Wilson v, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007500, A074832, A074833, A075235, A075236, A075237, A075238, A075239.

Prime[ Select[ Range[140], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ Prime[ # ], 4]], 4]] &]]

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A114018 Least n-digit prime whose digit reversal is also prime.

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A238852 Right-truncatable, reversible primes in base 100.

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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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A359136 Primes such that there is a nontrivial permutation which when applied to the digits produces a prime (Version 1).

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A033294 Squares which when written backwards remain square (final 0's excluded).

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