A007500 -id:A007500 - cp's OEIS frontend

A103172 Start of ten consecutive primes whose digit reversals are also prime.

1193, 91528739, 302706311, 777528457, 778286917, 924408493, 1177842077, 1477271183, 1477271249, 1801280717, 1811906567, 7060718569, 9338212141, 9387802769, 9387802807, 9387802817, 9427522387, 9427522409, 9944534927

Offset: 1

Labos Elemer, Jan 31 2005

			1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249 and 1259 are consecutive primes.
Their digit reversals, 3911, 1021, 3121, 7121, 3221, 9221, 1321, 7321, 9421 and 9521, are all prime.

Cf. A007500, A103169, A103170, A103171.

Edited and extended by David Wasserman, Sep 05 2006

Corrected by Farideh Firoozbakht, Sep 23 2009

A136634 Primes whose reversals in bases 10, 9, 8, 7, 6, 5, 4, 3 and 2 are all prime.

Original entry on oeis.org

93836531, 1819395637, 1919723027, 1963209431, 3277373311, 3540866053, 15969326033, 16075946743, 16735166477, 17145519379, 71606465171, 71624919101, 72338598089, 73544885809, 73939267019, 74592559721

Offset: 1

Harry J. Smith, Jan 15 2008

			Prime 93836531 reversed base 10 = 13563839, a prime.
93836531 = 215511462 base 9, reversed = 264115512 base 9 = 116986691, a prime.
93836531 = 545752363 base 8, reversed = 363257545 base 8 = 63790949, a prime.
93836531 = 2216411615 base 7, reversed = 5161146122 base 7 = 212620277, a prime.
93836531 = 13151124215 base 6, reversed = 51242115131 base 6 = 316991071, a prime.
93836531 = 143010232111 base 5, reversed = 111232010341 base 5 = 61594471, a prime.
93836531 = 11211331103303 base 4, reversed = 30330113311211 base 4 = 217152869, a prime.
93836531 = 20112120101112002 base 3, reversed = 20021110102121102 base 3 = 90058187, a prime.
93836531 = 101100101111101010011110011 base 2, reversed = 110011110010101111101001101 base 2 = 108617549, a prime.

Harry J. Smith, Table of n, a(n) for n = 1..45

Cf. A007500, A074832, A136186, A136187.

emirp(p,b)=my(q,t=p);while(t,q=b*q+t%b;t\=b);isprime(q) && p!=q
is(n)=for(b=2,10,if(!emirp(n,b),return(0))); isprime(n) \\ Charles R Greathouse IV, Sep 03 2013

A103169 Start of seven consecutive primes whose digit reversals are also prime.

Original entry on oeis.org

2, 727, 733, 1193, 1201, 1213, 1217, 11897, 18719, 79379, 125627, 334759, 334771, 743989, 910909, 920957, 928429, 941449, 1093571, 1215079, 1407181, 1466533, 1518863, 1648553, 1770829, 3170743, 3300593, 7321943, 7682687, 7755581

Offset: 1

Labos Elemer, Jan 31 2005

			1193, 1201, 1213, 1217, 1223, 1229 and 1231 are consecutive primes.
Their digit reversals, 3911, 1021, 3121, 7121, 3221, 9221 and 1321, are all prime.

Cf. A007500, A103170, A103171, A103172.

With[{s = Prime@ Range[10^6]}, IntegerReverse /@ Flatten@ Map[Take[#, Length@ # - 6] &, DeleteCases[SplitBy[Map[IntegerReverse, s], PrimeQ], k_ /; Or[CompositeQ@ First@ k, Length@ k < 7]]]] (* Michael De Vlieger, Jul 24 2017 *)

Edited by David Wasserman, Sep 05 2006

A359138 A359136 together with 2, 3, 5, 7.

Original entry on oeis.org

2, 3, 5, 7, 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

By including the "by convention" terms 2, 3, 5, and 7, many sequences such as A007500 are now subsequences.

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

Cf. A359136, A359137, A359139.

A046485 Sum of first n palindromic primes A002385.

Original entry on oeis.org

2, 5, 10, 17, 28, 129, 260, 411, 592, 783, 1096, 1449, 1822, 2205, 2932, 3689, 4476, 5273, 6192, 7121, 17422, 27923, 38524, 49835, 61246, 73667, 86388, 99209, 112540, 126371, 140302, 154643, 169384, 184835, 200386, 216447, 232808, 249369, 266030, 283501

Offset: 1

Patrick De Geest, Sep 15 1998

The subsequence of prime partial sum of palindromic primes begins: 2, 5, 17, 5273, 7121, 154643, 283501. What is the smallest nontrivial (i.e., multidigit) palindromic prime partial sum of palindromic primes? [Jonathan Vos Post, Feb 07 2010]

Vincenzo Librandi, Table of n, a(n) for n = 1..700
Patrick De Geest, World!Of Palindromic Primes

Cf. A002385, A007504, A046489.

Cf. A000040, A007500, A006567, A016041, A029732, A117697. [Jonathan Vos Post, Feb 07 2010]

t = {}; b = 10; Do[p = Prime[n]; i = IntegerDigits[p, b]; If[i == Reverse[i], AppendTo[t, p];(*Print[p.FromDigits[i]]*)], {n, 4000}]; Accumulate[t] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
Accumulate[Select[Prime[Range[10000]],IntegerDigits[#]==Reverse[ IntegerDigits[#]]&]] (* Harvey P. Dale, Aug 10 2013 *)

a(n) = Sum_{i=1..n} A002385(i) = Sum_{i=1..n} {p prime and R(p) = p, i.e., primes whose decimal expansion is a palindrome}. [Jonathan Vos Post, Feb 07 2010]

Offset set to 1 by R. J. Mathar, Feb 21 2010

A085300 a(n) is the least prime x such that when reversed it is a power of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 163, 18258901387, 90367894271, 13, 73, 1861, 344800741, 34351783286302805384336021, 940315563074788471, 1886172359328147919771, 14854831

Offset: 1

Labos Elemer, Jun 24 2003

A006567 (after rearranging terms) and A002385 are subsequences. - Chai Wah Wu, Jun 02 2016

			a(14)=344800741 means that 147008443=43^5=p(14)^5, where 5 is the smallest such exponent;
a(19) has 82 decimal digits and if reversed equals 39th power of p(19)=67.

Chai Wah Wu, Table of n, a(n) for n = 1..86

Cf. A085298, A057708, A058993, A056994, A059695, A003459, A007500, A055387, A061461, A068652.

from sympy import prime, isprime
def A085300(n):
    p = prime(n)
    q = p
    while True:
        m = int(str(q)[::-1])
        if isprime(m):
            return(m)
        q *= p # Chai Wah Wu, Jun 02 2016

A095180 Reverse digits of primes, append to sequence if result is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 13, 73, 17, 37, 97, 79, 101, 701, 311, 131, 941, 151, 751, 761, 971, 181, 191, 991, 113, 313, 733, 743, 353, 953, 373, 383, 983, 107, 907, 727, 337, 937, 347, 157, 757, 167, 967, 787, 797, 709, 919, 929, 739, 149, 359, 769, 179, 389, 199

Offset: 1

Cino Hilliard, Jun 21 2004

Conjecture: the Benford law limit is 2=Sum[N[Log[10, 1 + 1/d[[n]]]], {n, 1, Length[d]}]^2/(( #totalprimes/#totalPrimes)). At 50000 primes total it is 2.05931. - Roger L. Bagula and Gary W. Adamson, Jul 02 2008

Presumably this does not satisfy Benford's law. - N. J. A. Sloane, Feb 09 2017

			The prime 107 in reverse is 701 which is prime.

Indranil Ghosh, Table of n, a(n) for n = 1..25000 (terms 1..206 from Roger L. Bagula and Gary W. Adamson)
Eric Weisstein's World of Mathematics, Benford's Law.
Index entries for sequences related to Benford's law

Cf. A007500.

Cf. A004087, A010051.

a095180 n = a095180_list !! (n-1)
a095180_list =filter ((== 1) . a010051) a004087_list
-- Reinhard Zumkeller, Oct 14 2011

b = Flatten[Table[If[PrimeQ[Sum[IntegerDigits[Prime[n]][[i]]*10^(i - 1), {i, 1, Length[IntegerDigits[Prime[n]]]}]], Sum[IntegerDigits[Prime[n]][[i]]*10^(i - 1), {i, 1, Length[IntegerDigits[Prime[n]]]}], {}], {n, 1,1000}]] (* Roger L. Bagula and Gary W. Adamson, Jul 02 2008 *)
Select[FromDigits[Reverse[IntegerDigits[#]]]&/@Prime[Range[300]],PrimeQ] (* Harvey P. Dale, May 05 2015 *)

r(n) = forprime(x=1,n,y=eval(rev(x));if(isprime(y),print1(y","))) \ Get the reverse of the input string rev(str) = { local(tmp,j,s); tmp = Vec(Str(str)); s=""; forstep(j=length(tmp),1,-1, s=concat(s,tmp[j])); return(s) }

A103170 Start of eight consecutive primes whose digit reversals are also prime.

Original entry on oeis.org

727, 1193, 1201, 1213, 334759, 7904639, 7904651, 9094009, 9685771, 11875307, 12503017, 19776443, 32906869, 35414443, 37376201, 70252333, 71161309, 73694129, 77454067, 91528739, 91528777, 91528807, 93907523

Offset: 1

Labos Elemer, Jan 31 2005

			1193, 1201, 1213, 1217, 1223, 1229, 1231 and 1237 are consecutive primes.
Their digit reversals, 3911, 1021, 3121, 7121, 3221, 9221, 1321 and 7321, are all prime.

Cf. A007500, A103169, A103171, A103172.

Edited by David Wasserman, Sep 05 2006

A103171 Start of nine consecutive primes whose digit reversals are also prime.

Original entry on oeis.org

1193, 1201, 7904639, 91528739, 91528777, 120890249, 154984343, 174625597, 302706311, 302706319, 312700789, 318629783, 707262887, 756791029, 777528457, 777528461, 778286917, 778286947, 923780981, 924408493, 924408497, 958610069

Offset: 1

Labos Elemer, Jan 31 2005

There are large gaps in this sequence because all terms begin with 1, 3, 7, or 9.

			1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237 and 1249 are consecutive primes.
Their digit reversals, 3911, 1021, 3121, 7121, 3221, 9221, 1321, 7321 and 9421, are all prime.

Cf. A007500, A103169, A103170, A103172.

Edited by David Wasserman, Sep 05 2006

A178316 Primes whose digital rotation is still prime.

Original entry on oeis.org

2, 5, 11, 19, 61, 101, 109, 151, 181, 199, 601, 619, 659, 661, 1019, 1021, 1061, 1091, 1109, 1129, 1151, 1181, 1201, 1229, 1259, 1291, 1511, 1559, 1601, 1609, 1621, 1669, 1699, 1811, 1901, 1999, 6011, 6091, 6101, 6199, 6211, 6221, 6229, 6521, 6551, 6569

Offset: 1

David Nacin, May 24 2010

This means if written as in a digital clock and rotated 180 degrees around the center the result is also prime (possibly a different prime).

			For example 1259 becomes 6521 under such a rotation.

Guy, R. K., Unsolved Problems in Number Theory, p 15 This sequence is related to the palindromic primes with symmetries as in Guy's book.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007500, A055387, A018847.

Select[Range[6570],PrimeQ[#]&&PrimeQ[FromDigits[Reverse[IntegerDigits[#]/.{6->9,9->6}]]]&&ContainsOnly[IntegerDigits[#],{0,1,2,5,6,8,9}]&] (* James C. McMahon, Apr 09 2024 *)

from itertools import count, islice, product
from sympy import isprime
def A178316_gen():
    yield from (2,5)
    r = ''.maketrans('69','96')
    for l in count(1):
        for a in '125689':
            for d in product('0125689',repeat=l):
                s = a+''.join(d)
                m = int(s)
                if isprime(m) and isprime(int(s[::-1].translate(r))):
                    yield m
A178316_list = list(islice(A178316_gen(),40)) # Chai Wah Wu, Apr 09 2024

cp's OEIS Frontend

A103172 Start of ten consecutive primes whose digit reversals are also prime.

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A136634 Primes whose reversals in bases 10, 9, 8, 7, 6, 5, 4, 3 and 2 are all prime.

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A103169 Start of seven consecutive primes whose digit reversals are also prime.

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A359138 A359136 together with 2, 3, 5, 7.

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A046485 Sum of first n palindromic primes A002385.

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A085300 a(n) is the least prime x such that when reversed it is a power of prime(n).

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A095180 Reverse digits of primes, append to sequence if result is a prime.

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A103170 Start of eight consecutive primes whose digit reversals are also prime.

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A103171 Start of nine consecutive primes whose digit reversals are also prime.

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