A006567 -id:A006567 - cp's OEIS frontend

A155512 Emirps with digits 0 and 1 only.

10011101, 10111001, 100100111, 111001001, 1000110101, 1001001011, 1010001101, 1010011111, 1010110001, 1011000101, 1101001001, 1111100101, 10010100101, 10100000011, 10100000111, 10100101001, 10111001011, 11000000101

Offset: 1

Lekraj Beedassy, Jan 23 2009

Subsequence of A128390 and of A020449.

Harvey P. Dale, Table of n, a(n) for n = 1..3000

A006567, A155513, A155514

emQ[ls_List]:=Module[{rev=Reverse[ls]},Length[ls]==Length[rev]&& ls!=rev && PrimeQ[FromDigits[ls]]&&PrimeQ[FromDigits[rev]]]; Union[Flatten[ Table[FromDigits/@Select[Tuples[{1,0},n],emQ],{n,11}]]] (* Harvey P. Dale, Nov 30 2011 *)

First two missed entries included. Lekraj Beedassy, May 30 2009

More terms from Sean A. Irvine, Mar 04 2010

A176371 Primes p such that reversal(p) - 13 is a square.

Original entry on oeis.org

31, 41, 71, 83, 281, 311, 431, 479, 733, 751, 797, 2011, 2857, 3163, 4373, 4397, 4943, 7541, 7577, 7583, 9413, 9491, 20533, 20731, 20771, 24151, 24547, 24767, 26249, 28979, 31121, 41201, 41609, 43321, 43391, 43753, 45641, 49459, 49463, 49811, 49891

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 16 2010

R(n) denotes the Reversal of a natural number n
List of all (p,N) for p < 10^6 - 1:
(*) for emirp pair (p,R(p)), (+) if square base N is a prime
(41,1), (71,2) (+) (*), (83,5) (+), (281,13) (+), (311,10) (*), (431,11) (+), (479,31) (+), (733,18) (*), (751,12) (*), (797,28),
(2011,33), (2857,87), (4373,61) (+), (4397,89) (+), (4943,59) (+), (7541,38), (7577,88) (*), (7583,62), (9413,56), (9491,44) (*), (20533,183), (20731,117), (20771,133), (24151,123), (24547,273), (24767,277) (+), (26249,307) (+), (28979,313) (+), (31121,110) (*), (41201,101) (+),
(41609,301), (43321,111), (43391,139) (+), (43753,189), (45641,121), (49459,309), (49463,191) (+), (49811,109), (49891,141), (71293,198) (*),
(73133,182), (73471,132), (73597,282) (*), (75521,112), (77611,108) (*), (77849,308), (77863,192) (*), (79613,178), (79841,122) (*), (83207,265),
(83231,115), (83243,185), (83299,315), (90031,114) (*), (92801,104), (96431,116) (*), (98057,274)

			41 = prime(13), R(41) - 13 = 14 - 13 = 1^2, is a term.
71 = prime(20), 17 - 13 = 2^2, is a term.
83 = prime(23), 38 - 13 = 5^2, is a term.
797 = prime(139) = palindromic prime(18), N = 28^2, is also a term.
Note successive terms that are also consecutive primes: p(17) = 7577, p(18) = 7583, p(36) = 49459, p(37) = 49463, p(46) = 77849, p(47) = 77863.

W. W. R. Ball, H. S. M.Coxeter: Mathematical Recreations and Essays, Dover Publications, 13th edition, 1987
O. Fritsche, R. Mischak and T. Krome: Verflixt und zugeknobelt, Mehr mathematische Raetselgeschichten, Rowohlt TB. Nr.62190, 2007
C. W. Trigg, Primes with Reverses That Are Powers, J. Rec. Math. 17, 1985

Chai Wah Wu, Table of n, a(n) for n = 1..7605

Cf. A000040, A000290, A002385, A006567, A007488.

isok(n) = {if (! isprime(n), return (0)); d = digits(n); revn = sum(i=1, #d, d[i]*10^(i - 1)); issquare(revn-13);} \\ Michel Marcus, Aug 25 2013

from sympy import isprime
A176371_list, i, j = [], 0, 13
while j < 10**10:
    p = int(str(j)[::-1])
    if j % 10 and isprime(p):
        A176371_list.append(p)
    j += 2*i+1
    i += 1
A176371_list = sorted(A176371_list) # Chai Wah Wu, Dec 17 2015

Two more terms 31 and 3163 added by Michel Marcus, Aug 25 2013

A210498 Prime numbers that become emirps when their least-significant-digit is deleted.

Original entry on oeis.org

131, 137, 139, 173, 179, 311, 313, 317, 373, 379, 719, 733, 739, 797, 971, 977, 1493, 1499, 1571, 1579, 1993, 1997, 1999, 3119, 3371, 3373, 3593, 7013, 7019, 7331, 7333, 7393, 7433, 7517, 7691, 7699, 9371, 9377, 9413, 9419, 9533, 9539, 9677, 9679, 9719, 9833

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 03 2013

This sequence (like the emirps) experiences large gaps when the most-significant-digit is {2,4,5,6,8}.

			Example: a(1)=131, which becomes 13 upon deletion, which is an emirp.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A006567, A024770, A000040, A225235

library(gmp); isemirp<-function(x) isprime(x) & (j=paste(rev(unlist(strsplit(as.character(x),split=""))),collapse=""))!=x & isprime(j);i=as.bigz(0); y=as.bigz(rep(0,100)); len=0;
while(len<100)
    if(isprime((i=nextprime(i))))
        if(isemirp(as.bigz(substr(i,1,nchar(as.character(i))-1))))
            y[(len=len+1)]=i

A225235 Emirps whose internal digits are also an emirp.

Original entry on oeis.org

1979, 3319, 3371, 3373, 3719, 3733, 7177, 7717, 9133, 9173, 9791, 10177, 10711, 10739, 11071, 11497, 11579, 11677, 13477, 13591, 13597, 17011, 17393, 17519, 19531, 19913, 30139, 30319, 30971, 31139, 31799, 31991, 37619, 39371, 39419, 39839, 70313, 70373, 70717

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 03 2013

This sequence (like the emirps) experiences large gaps when the most-significant-digit is {2,4,5,6,8}.

			7177 and 10177 are in the sequence because both are emirps, and both become the emirp 17 upon deletion of their first and last digits.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A210498, A006567, A069686, A210498.

library(gmp); isemirp<-function(x) isprime(x) & (j=paste(rev(unlist(strsplit(as.character(x), split=""))), collapse=""))!=x & isprime(j);
no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s};
i=as.bigz(0); y=as.bigz(rep(0, 100)); len=0;
while(len<100) if(isemirp((i=nextprime(i)))) if(isemirp(as.bigz(no0(substr(i,2,nchar(as.character(i))-1))))) y[(len=len+1)]=i;
as.vector(y)

A346026 Primes that are the first in a run of exactly 6 emirps.

Original entry on oeis.org

10039, 14891, 39791, 119773, 149561, 162293, 163781, 176903, 181751, 197383, 336689, 392911, 393361, 714361, 715361, 779003, 971141, 995443, 996539, 1165037, 1284487, 1307729, 1447151, 1611877, 1640539, 1789621, 1891147, 3136909, 3150557, 3284447, 3339943

Offset: 1

Lars Blomberg, Jul 14 2021

There are large gaps in this sequence because all terms need to begin with 1, 3, 7, or 9 otherwise the reversal is composite.

			a(1) = 10039 because of the eight consecutive primes 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093 all except 10037 and 10093 are emirps and this is the first such occurrence.

Subsequence of A006567 (emirps).

Cf. A003684, A048052, A048054, A071612, A346022, A346023, A346024, A346025.

EmQ[n_]:=(s=IntegerReverse@n;PrimeQ@s&&n!=s);
Select[Prime@Range[2,50000],Boole[EmQ/@NextPrime[#,Range[-1,6]]]=={0,1,1,1,1,1,1,0}&] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

from sympy import isprime, nextprime, prime, primerange
def isemirp(p): s = str(p); return s != s[::-1] and isprime(int(s[::-1]))
def aupto(limit, runlength=6):
  alst = []
  pvec = list(primerange(1, prime(runlength+2)+1))
  evec = [int(isemirp(p)) for p in pvec]
  target = [0] + [1 for i in range(runlength)] + [0]
  p = nextprime(pvec[-1])
  while pvec[1] <= limit:
    if evec == target: alst.append(pvec[1])
    pvec = pvec[1:] + [p]; evec = evec[1:] + [isemirp(p)]; p = nextprime(p)
    strp = str(p)
    if strp[0] in "24568": # skip large gaps (p is a prime, not an emirp)
      evec = [0 for i in range(runlength+2)]
      pvec = [0 for i in range(runlength+2)]
      p = nextprime(int(str(int(strp[0])+1)+'0'*(len(strp)-1)))
  return alst
print(aupto(3339943)) # Michael S. Branicky, Jul 14 2021

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

A054218 Palindromic primes of the form 'primemirp' resulting from A054217.

Original entry on oeis.org

2, 3, 5, 7, 131, 313, 373, 797, 11311, 17971, 18181, 19991, 35353, 72727, 78787, 90709, 93739, 96769, 98389, 1153511, 1193911, 1201021, 1409041, 1583851, 1597951, 1657561, 1831381, 1879781, 3083803, 3089803, 3319133, 3343433, 3391933, 3541453, 3643463

Offset: 1

Patrick De Geest, Feb 15 2000

Original idea from G. L. Honaker, Jr.

			Prime 113 has emirp 311 and 11311 is a palindromic prime.

Peter Rowlett, Table of n, a(n) for n = 1..8668

Cf. A054217, A000040, A006567, A048054, A002385.

from sympy import isprime
for i in range(2,10**7):
    if isprime(i):
        emirp = int(str(i)[-1::-1])
        if isprime(emirp):
            primemirp = int(str(i)+str(emirp)[1:])
            if isprime(primemirp):
                print(primemirp)
# Peter Rowlett, Nov 16 2023

a(33)-a(35) from Peter Rowlett, Nov 16 2023

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

A155507 Emirps with digits 1 and 9 only.

Original entry on oeis.org

199, 991, 91199, 99119, 111119, 111919, 119191, 191911, 911111, 919111, 991999, 999199, 1191119, 1191191, 1191991, 1911911, 1991911, 9111911, 11111911, 11191991, 11911111, 11919991, 19111991, 19911191, 19919111, 19991911

Offset: 1

Lekraj Beedassy, Jan 23 2009

Subsequence of A128390.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

A006567, A128390, A155508, A155509

emrpQ[n_]:=Module[{r=IntegerReverse[n]},r!=n&&AllTrue[{r,n},PrimeQ]]; Table[Select[FromDigits/@Tuples[{1,9},n],emrpQ],{n,8}]//Flatten (* Harvey P. Dale, Aug 09 2017 *)

More terms from Sean A. Irvine, Apr 05 2010

A155508 Lesser of emirps (pairs) with digits 1 and 9 only.

Original entry on oeis.org

199, 91199, 111119, 111919, 119191, 991999, 1191119, 1191191, 1191991, 11111911, 11191991, 11919991, 19111991, 91111199, 111111199, 111199199, 191111119, 199991999, 919919999, 1111111999, 1191111991, 1199111191, 1911199999

Offset: 1

Lekraj Beedassy, Jan 23 2009

Harvey P. Dale, Table of n, a(n) for n = 1..660

A006567, A155507, A155509

Flatten[Table[Select[Subsets[Select[FromDigits/@Tuples[{1,9},n],PrimeQ],{2}],AllTrue[#,PrimeQ]&&#[[1]]==IntegerReverse[#[[2]]]&][[All,1]],{n,3,10}]/.{}->Nothing] (* Harvey P. Dale, Aug 21 2020 *)

More terms from Sean A. Irvine, Apr 06 2010

cp's OEIS Frontend

A155512 Emirps with digits 0 and 1 only.

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A176371 Primes p such that reversal(p) - 13 is a square.

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A210498 Prime numbers that become emirps when their least-significant-digit is deleted.

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A225235 Emirps whose internal digits are also an emirp.

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A346026 Primes that are the first in a run of exactly 6 emirps.

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A046485 Sum of first n palindromic primes A002385.

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A054218 Palindromic primes of the form 'primemirp' resulting from A054217.

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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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