A048054 -id:A048054 - cp's OEIS frontend

A346024 Primes that are the first in a run of exactly 4 emirps.

733, 7177, 9011, 11551, 11777, 12107, 13147, 13259, 13693, 14563, 19219, 19531, 19661, 31891, 32467, 35117, 35311, 36097, 36187, 38351, 38903, 70241, 70921, 75721, 77323, 78607, 79399, 79531, 90121, 91183, 92297, 92479, 92959, 93581, 94121, 95111, 95791, 96857

Offset: 1

Lars Blomberg, Jul 02 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) = 733 because of the six consecutive primes 727, 733, 739, 743, 751, 757 all except 727 and 757 are emirps and this is the first such occurrence.

Subsequence of A006567 (emirps)

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

Select[Prime@Range@10000,Boole[PrimeQ@#&&!PalindromeQ@#&/@(IntegerReverse/@NextPrime[#,Range[-1,4]])]=={0,1,1,1,1,0}&] (* Giorgos Kalogeropoulos, Jul 04 2021 *)

from sympy import isprime, primerange
def isemirp(p): s = str(p); return s != s[::-1] and isprime(int(s[::-1]))
def aupto(limit):
    alst, pvec, evec = [], [2, 3, 5, 7, 11, 13], [0, 0, 0, 0, 0, 0]
    for p in primerange(17, limit+1):
        if evec == [0, 1, 1, 1, 1, 0]: alst.append(pvec[1])
        pvec = pvec[1:] + [p]; evec = evec[1:] + [isemirp(p)]
    return alst
print(aupto(97000)) # Michael S. Branicky, Jul 04 2021

A346025 Primes that are the first in a run of exactly 5 emirps.

Original entry on oeis.org

3371, 9769, 11699, 11953, 15493, 34549, 72307, 72547, 105653, 106391, 109849, 129587, 139387, 144407, 169067, 170759, 178333, 193261, 193877, 316073, 324031, 324893, 325163, 333923, 339671, 375787, 381859, 389287, 701383, 701593, 712289, 722633, 744377, 777349

Offset: 1

Lars Blomberg, Jul 02 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) = 3371 because of the seven consecutive primes 3361, 3371, 3373, 3389, 3391, 3407, 3413 all except 3361 and 3413 are emirps and this is the first such occurrence.

Subsequence of A006567 (emirps).

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

Select[Prime@Range@20000,Boole[PrimeQ@#&&!PalindromeQ@#&/@(IntegerReverse/@NextPrime[#,Range[-1,5]])]=={0,1,1,1,1,1,0}&] (* Giorgos Kalogeropoulos, Jul 04 2021 *)

from sympy import isprime, primerange
def isemirp(p): s = str(p); return s != s[::-1] and isprime(int(s[::-1]))
def aupto(limit):
    alst, pvec, evec = [], [2, 3, 5, 7, 11, 13, 17], [0, 0, 0, 0, 0, 0, 0]
    for p in primerange(19, limit+1):
        if evec == [0, 1, 1, 1, 1, 1, 0]: alst.append(pvec[1])
        pvec = pvec[1:] + [p]; evec = evec[1:] + [isemirp(p)]
    return alst
print(aupto(780000)) # Michael S. Branicky, Jul 04 2021

A054217 Primes p with property that p concatenated with its emirp p' (prime reversal) forms a palindromic prime of the form 'primemirp' (rightmost digit of p and leftmost digit of p' are blended together - p and p' palindromic allowed).

Original entry on oeis.org

2, 3, 5, 7, 13, 31, 37, 79, 113, 179, 181, 199, 353, 727, 787, 907, 937, 967, 983, 1153, 1193, 1201, 1409, 1583, 1597, 1657, 1831, 1879, 3083, 3089, 3319, 3343, 3391, 3541, 3643, 3853, 7057, 7177, 7507, 7681, 7867, 7949, 9103, 9127, 9173, 9209, 9439, 9547, 9601

Offset: 1

Patrick De Geest, Feb 15 2000

Original idea from G. L. Honaker, Jr..

			E.g., prime 113 has emirp 311 and 11311 is a palindromic prime, so 113 is a term.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A054218, A000040, A006567, A048054, A002385.

empQ[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];And@@PrimeQ[ {FromDigits[ rev],FromDigits[Join[Most[idn],rev]]}]]; Select[Prime[ Range[ 1200]],empQ] (* Harvey P. Dale, Mar 26 2013 *)

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n); srev = s[::-1]
    return isprime(int(srev)) and isprime(int(s[:-1] + srev))
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Nov 17 2023

Corrected (a(30)=3089 inserted) by Harvey P. Dale, Mar 26 2013

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

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

A346027 Primes that are the first in a run of exactly 7 emirps.

Original entry on oeis.org

11897, 18719, 125627, 743989, 910909, 920957, 928429, 941449, 1093571, 1407181, 1466533, 1518863, 1648553, 1770829, 3170743, 3300593, 7321943, 7682687, 7755581, 9013351, 12890047, 13267459, 14113199, 16413013, 16944341, 17316031, 18447001, 18490267, 18964111

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) = 11897 because of the nine consecutive primes 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941 all except 11887 and 11941 are emirps and this is the first such occurrence.

Subsequence of A006567 (emirps).

Cf. A003684, A048052, A048054, A071612, A346022, A346023, A346024, A346025, A346026.

EmQ[n_]:=(s=IntegerReverse@n;PrimeQ@s&&n!=s);
Monitor[Do[p=Prime@k;If[MemberQ[{1,3,7,9},First@IntegerDigits@p],If[Boole[EmQ/@NextPrime[p,Range[-1,7]]]=={0,1,1,1,1,1,1,1,0},Print@p]],{k,10^6}],p] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

# uses code in A346026
print(aupto(10**7, runlength=7)) # Michael S. Branicky, Jul 14 2021

A346028 Primes that are the first in a run of exactly 8 emirps.

Original entry on oeis.org

334759, 9094009, 9685771, 11875307, 12503017, 19776443, 32906869, 35414443, 37376201, 70252333, 71161309, 73694129, 77454067, 93907523, 98606489, 100545637, 104827991, 112604857, 147009703, 155376791, 183766217, 187717499, 194024953, 196702423, 314716411

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) = 334759 because of the 10 consecutive primes 334753, 334759, 334771, 334777, 334783, 334787, 334793, 334843, 334861, 334877 all except 334753 and 334877 are emirps and this is the first such occurrence.

Subsequence of A006567 (emirps).

Cf. A003684, A048052, A048054, A071612, A346022, A346023, A346024, A346025, A346026, A346027.

EmQ[n_]:=(s=IntegerReverse@n;PrimeQ@s&&n!=s);
Monitor[Do[p=Prime@k;If[MemberQ[{1,3,7,9},First@IntegerDigits@p],If[Boole[EmQ/@NextPrime[p,Range[-1,8]]]=={0,1,1,1,1,1,1,1,1,0},Print@p]],{k,10^6}],p] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

# uses code in A346026
print(aupto(10**7, runlength=8)) # Michael S. Branicky, Jul 14 2021

A178545 Primes p such that q = p^2 + p + 1 is an emirp.

Original entry on oeis.org

3, 5, 41, 59, 839, 857, 1811, 1931, 3011, 3221, 3407, 3671, 8387, 8543, 8627, 9719, 9743, 9803, 10781, 11549, 12647, 13469, 13487, 13499, 13613, 13931, 14087, 17477, 17573, 17837, 18089, 18269, 19319, 19403, 19661, 19991, 27191, 27947, 31223, 33311, 34313

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 29 2010

It is conjectured (but still an open problem) that there exist infinitely many primes of the form n^2 + n + 1 = ((2*n+1)^2 + 3)/4.
Landau's 4th problem from (1912, 5th Congress of Mathematicians in Cambridge) conjectures that there are infinitely many primes of the form n^2 + 1 (also Euler 1760; Mirsky 1949).
Hardy and Littlewood proposed a conjecture about the asymptotic number of primes of the form n^2 + 1.
An emirp ("prime" spelled backwards) is a prime whose reversal is a different prime, the reversal of q is denoted by R(q).
It is conjectured but also unproved that there are infinitely many emirps (see A048054).
For p > 3 necessarily p of the form 6*k + 5 as (6*k+1)^2 + (6*k+1) + 1 a multiple of 3.

			3^2 + 3 + 1 = 13 = prime(6), R(13) = prime(11), 3 is first term.
5^2 + 5 + 1 = 31 = prime(11), R(31) = prime(6), 5 is 2nd term.
q = 1811^2 + 1811 + 1 = 3281533 = prime(235691), R(q) = prime(240351), first case that p = 1811 = prime(280) = emirp(87) is itself an emirp.

M. Gardner: Die magischen Zahlen des Dr. Matrix, Krueger Verlag, Frankfurt am Main, 1987
R. Guy: Unsolved Problems in Number Theory,3rd edition, Springer, New York, 2004
G. H. Hardy, E. M. Wright: Einfuehrung in die Zahlentheorie, R. Oldenburg, Muenchen, 1958

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A002383, A048054, A006567, A109308, A109309.

filter:= proc(p) local q,qr;
   if not isprime(p) then return false fi;
   q:= p^2+p+1;
   if not isprime(q) then return false fi;
   qr:= revdigs(q);
   qr <> q and isprime(qr);
end proc:
select(filter, [3,seq(i,i=5..50000,6)]); # Robert Israel, Dec 04 2016

EmirpQ[n_] := If[ PrimeQ@n, Block[{id = IntegerDigits@n}, rid = Reverse@ id; rid != id && PrimeQ@ FromDigits@ rid]]; Select[ Prime@ Range@ 3700, EmirpQ[ #^2 + # + 1] &] (* Robert G. Wilson v, Jul 26 2010 *)
p2emrpQ[p_]:=With[{q=p^2+p+1},!PalindromeQ[q]&&AllTrue[{q,IntegerReverse[q]},PrimeQ]]; Select[Prime[Range[3700]],p2emrpQ] (* Harvey P. Dale, Mar 10 2025 *)

More terms from Robert G. Wilson v, Jul 26 2010

A346029 Primes that are the first in a run of exactly 9 emirps.

Original entry on oeis.org

7904639, 120890249, 154984343, 174625597, 312700789, 318629783, 707262887, 756791029, 923780981, 958610069, 1049344897, 1068171977, 1117675201, 1194919381, 1327765591, 1368391847, 1385828243, 1846629391, 1976590081, 3117896521, 3182618969, 3322051367

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) = 7904639 because of the 11 consecutive primes 7904629, 7904639, 7904651, 7904653, 7904657, 7904669, 7904683, 7904707, 7904719, 7904723, 7904731 all except 7904629 and 7904731 are emirps and this is the first such occurrence.

Subsequence of A006567 (emirps).

Cf. A003684, A048052, A048054, A071612, A346022, A346023, A346024, A346025, A346026, A346027, A346028.

EmQ[n_]:=(s=IntegerReverse@n;PrimeQ@s&&n!=s);
Monitor[Do[p=Prime@k;If[MemberQ[{1,3,7,9},First@IntegerDigits@p],If[Boole[EmQ/@NextPrime[p,Range[-1,9]]]==Join[{0},1~Table~9,{0}],Print@p]],{k,10^8}],p] (* Giorgos Kalogeropoulos, Jul 27 2021 *)

# uses code in A346026
print(aupto(10**7, runlength=9)) # Michael S. Branicky, Jul 14 2021

A366910 Number of n-bit binary reversible primes.

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 9, 14, 27, 36, 69, 94, 178, 308, 589, 908, 1540, 2814, 5158, 9210, 16732, 29392, 55109, 101120, 179654, 332130, 625928, 1136814, 2120399, 3963166, 7377931, 13878622, 25958590, 48421044, 92163237, 173672988, 325098134, 617741968, 1177573074, 2221353224, 4222570054

Offset: 1

Cathy Swaenepoel, Oct 27 2023

Number of primes p in [2^(n-1),2^n) whose reverse in base 2 is also prime. The count includes palindromic primes in base 2.

			The 5-bit binary reversible primes are 17="10001", 23="10111", 29="11101" and 31="11111", so a(5)=4.

Cathy Swaenepoel, Table of n, a(n) for n = 1..50
Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 34.

Cf. A074831, A074832, A117773, A048054.

cp's OEIS Frontend

A346024 Primes that are the first in a run of exactly 4 emirps.

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A346025 Primes that are the first in a run of exactly 5 emirps.

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A054217 Primes p with property that p concatenated with its emirp p' (prime reversal) forms a palindromic prime of the form 'primemirp' (rightmost digit of p and leftmost digit of p' are blended together - p and p' palindromic allowed).

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

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

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A346027 Primes that are the first in a run of exactly 7 emirps.

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A346028 Primes that are the first in a run of exactly 8 emirps.

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A178545 Primes p such that q = p^2 + p + 1 is an emirp.

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