A006567 -id:A006567 - cp's OEIS frontend

A155513 Lesser of emirps (pairs) with digits 0 and 1 only.

10011101, 100100111, 1000110101, 1001001011, 1010001101, 1010011111, 10010100101, 10100000011, 10100000111, 10111001011, 100000010101, 100000011011, 100101110111, 100101111001, 100110101011, 100110101111

Offset: 1

Lekraj Beedassy, Jan 23 2009

A006567, A155512, A155514

First missed entry added. Lekraj Beedassy, May 30 2009

More terms from Sean A. Irvine, Mar 04 2010

A156793 Indices of primes whose reversal is a different prime.

Original entry on oeis.org

6, 7, 11, 12, 20, 21, 22, 25, 28, 30, 35, 37, 39, 41, 46, 64, 68, 69, 72, 77, 126, 127, 130, 131, 132, 133, 135, 136, 155, 159, 160, 162, 163, 164, 166, 167, 169, 172, 173, 174, 178, 180, 182, 184, 185, 186, 190, 191, 194, 196, 197, 198, 199, 200, 201, 202, 203

Offset: 1

Juri-Stepan Gerasimov, Feb 16 2009

Indices of primes in A006567.

			If prime(n=6)=13(1<3) and 31=prime, then 6=a(1). If prime(n=7)=17(1<7) and 71=prime, then 7=a(2). If prime(n=11)=31(3>1) and 13=prime, then 11=a(3), etc.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006567.

difpQ[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn];idn!=ridn && PrimeQ[FromDigits[ridn]]]; PrimePi/@ Select[ Prime[ Range[ 250]], difpQ] (* Harvey P. Dale, Nov 23 2011 *)

45 replaced by 46. 127, 131, 136, 155, 159 inserted, etc. - R. J. Mathar, Feb 26 2009

A168159 Distance of the least reversible n-digit prime from 10^(n-1).

Original entry on oeis.org

1, 1, 1, 9, 7, 49, 33, 169, 7, 7, 207, 237, 91, 313, 261, 273, 79, 49, 2901, 51, 441, 193, 9, 531, 289, 1141, 67, 909, 331, 753, 2613, 657, 49, 4459, 603, 1531, 849, 2049, 259, 649, 2119, 1483, 63, 6747, 519, 3133, 937, 1159, 1999, 6921, 2949, 613, 4137, 1977, 31

Offset: 1

M. F. Hasler, Nov 21 2009

A (much) more compact form of A114018 (cf. formula). Since this sequence and A114018 refer to "reversible primes" (A007500), while A122490 seems to use "emirps" (A006567), a(n+1) differs from A122490(n) iff 10^n+1 is prime <=> a(n+1)=1 <=> A114018(n)=10^n+1.

Michael S. Branicky, Table of n, a(n) for n = 1..500

Table[p = NextPrime[y = 10^(n - 1)]; While[! PrimeQ[FromDigits[Reverse[IntegerDigits[p]]]], p = NextPrime[p]]; p - y, {n, 55}] (* Jayanta Basu, Aug 09 2013 *)

for(x=1,1e99, until( isprime(x=nextprime(x+1)) & isprime(eval(concat(vecextract(Vec(Str(x)),"-1..1")))),);print1(x-10^ (#Str(x)-1),", "); x=10^#Str(x)-1)

from sympy import isprime
def c(n): return isprime(n) and isprime(int(str(n)[::-1]))
def a(n): return next(p-10**(n-1) for p in range(10**(n-1), 10**n) if c(p))
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Jun 27 2022

a(n)=A114018(n)-10^(n-1)

A173594 Near-repdigit emirps.

Original entry on oeis.org

113, 199, 311, 337, 733, 991, 1151, 1181, 1511, 1811, 3343, 3373, 3433, 3733, 7177, 7577, 7717, 7757, 11161, 16111, 77797, 79777, 98999, 99989, 111119, 111211, 112111, 323333, 333323, 333337, 333433, 334333, 733333, 777787, 777877, 778777, 787777, 911111

Offset: 1

Lekraj Beedassy, Feb 22 2010

Entries of A164937 that are emirps (A006567), i.e., A164937 INTERSECTION A006567.

Cf. A164937, A173595, A173596

Select[Prime[Range[26,75000]],!PalindromeQ[#]&&PrimeQ[IntegerReverse[ #]] && Count[ DigitCount[#],0]==8&&MemberQ[DigitCount[#],1]&] (* Harvey P. Dale, Sep 17 2021 *)

A175215 The smaller member of a twin prime pair in which both primes are emirps.

Original entry on oeis.org

71, 1031, 1151, 1229, 3299, 3371, 3389, 3467, 3851, 7457, 7949, 9011, 9437, 10007, 10067, 10457, 10889, 11159, 11699, 11717, 11777, 11969, 12071, 12107, 13709, 13757, 14447, 14549, 14591, 15731, 16451, 17207, 17681, 17747, 17909, 18911, 19421, 19541

Offset: 1

Lekraj Beedassy, Mar 06 2010

Subsequence of A001359 and of A006567.

Metin Sariyar, Table of n, a(n) for n = 1..16451
Green, Prime Curios, Entry 71
Carlos Rivera, Puzzle 973. Largest known twin & emirp, The Prime Puzzles & Problems Connection.

Cf. A001359, A006567.

emirp:=func; [p:p in PrimesUpTo(20000)| emirp(p) and emirp(p+2)]; // Marius A. Burtea, Dec 17 2019

read("transforms") ; isA001359 := proc(n) isprime(n) and isprime(n+2) ; end proc:
isA006567 := proc(n) local r ; r := digrev(n) ; isprime(n) and isprime(r) and n<> r ; end proc:
isA175215 := proc(n) isA001359(n) and isA006567(n) and isA006567(n+2) ; end proc:
for i from 1 to 10000 do p := ithprime(i) ; if isA175215(p) then printf("%d,",p) ; end if; end do: # R. J. Mathar, Mar 16 2010

Do[IR=IntegerReverse;Q=PrimeQ;If[Q[n]&&Q[n+2]&&Q[IR[n]]&&PrimeQ[IR[n+2]]&&!n==IR[n]&&!(n+2)==IR[n+2],Print[n]],{n,5,10^5,6}] (* Metin Sariyar, Dec 17 2019 *)

11699 inserted, 14921 -> 19421 corrected by R. J. Mathar, Mar 16 2010

A179032 Emirps whose only prime digits are one or more 2's.

Original entry on oeis.org

1021, 1201, 1229, 1249, 1429, 9029, 9209, 9221, 9241, 9421, 10429, 11621, 12109, 12119, 12149, 12241, 12269, 12289, 12491, 12611, 12619, 12641, 12689, 12809, 12829, 12841, 12919, 14029, 14221, 14621, 14629, 14821, 14929, 16249, 16829

Offset: 1

Lekraj Beedassy, Jun 25 2010

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006567, A179033.

Select[Prime[Range[2000]],!PalindromeQ[#]&&PrimeQ[IntegerReverse[#]] && Union[ Select[ IntegerDigits[ #],PrimeQ]]=={2}&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 06 2020 *)

Terms confirmed by Ray Chandler, Jul 13 2010

A179033 Emirps with a single 2 as the only prime digit.

Original entry on oeis.org

1021, 1201, 1249, 1429, 9029, 9209, 9241, 9421, 10429, 11621, 12109, 12119, 12149, 12491, 12611, 12619, 12641, 12689, 12809, 12841, 12919, 14029, 14621, 14629, 14821, 14929, 16249, 16829, 18269, 19219, 19249, 19421, 90121, 90821, 91121

Offset: 1

Lekraj Beedassy, Jun 25 2010

			Note that 2 and 929 are not emirps because they are palindromes.

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

Cf. A006567, A179032.

Dmax:= 6: # to get all terms with up to Dmax digits
Res:= NULL:
npd:= [0,1,4,6,8,9]:
for dd from 3 to Dmax do
  R:= [seq(seq([seq(npd[j+1],j=convert(6*x+j,base,6))],
    x=[$6^(dd-3) .. 2*6^(dd-3)-1, $5*6^(dd-3)..6^(dd-2)-1]),j=[1,5])];
  for p from 2 to dd-1 do
    for r in R do
      x:= [op(r[1..p-1]),2,op(r[p..-1])];
      v1:= add(x[i]*10^(i-1),i=1..dd);
      v2:= add(x[-i]*10^(i-1),i=1..dd);
      if v1 < v2 and isprime(v1) and isprime(v2) then Res:= Res,v1,v2; if min(v1,v2) < 10^3  then print(dd,p,r,x,v1,v2) fi fi
od od od:
sort([Res]); # Robert Israel, Jun 02 2016

emrp[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];PrimeQ[FromDigits[rev]]&&rev!=idn]
only2[n_]:=DigitCount[n,10,{3,5,7}]=={0,0,0}&&DigitCount[n,10,2]==1
Select[Select[Prime[Range[10000]],emrp],only2] (* Harvey P. Dale, Jan 22 2011 *)

Terms confirmed by Ray Chandler, Jul 13 2010

Definition improved by Harvey P. Dale, Jul 17 2010

A179034 Emirps whose only prime digits are 3's.

Original entry on oeis.org

13, 31, 113, 311, 389, 983, 1031, 1033, 1103, 1193, 1301, 1381, 1399, 1439, 1831, 1913, 1933, 3011, 3019, 3049, 3083, 3089, 3109, 3163, 3169, 3191, 3301, 3319, 3343, 3389, 3391, 3433, 3463, 3469, 3613, 3643, 3803, 3889, 3911, 9013, 9103, 9133, 9341

Offset: 1

Lekraj Beedassy, Jun 25 2010

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006567, A179035.

em3Q[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn]; idn!= ridn&&PrimeQ[FromDigits[ridn] && Count[idn,2] == Count[idn,5] == Count[idn,7] == 0&&Count[idn,3]>0]]; Select[Prime[Range[1200]],em3Q] (* Harvey P. Dale, Nov 30 2012 *)

Corrected by Ray Chandler, Jul 13 2010

A182721 Mountain emirps.

Original entry on oeis.org

1231, 1321, 1381, 1471, 1741, 1831, 12491, 12641, 12841, 13591, 13751, 13781, 13841, 14591, 14621, 14821, 14831, 14891, 15731, 15791, 18731, 19421, 19531, 19541, 19751, 19841, 123731, 123821, 124951, 124981, 125641, 125651, 125791, 125821, 125941, 126761, 126851

Offset: 1

Omar E. Pol, Dec 21 2010

Intersection of emirps A006567 and mountain numbers A134941.
The smallest mountain emirp 1231 and other terms of this sequence was mentioned by Loungrides in Prime Curios! (see link).
Question: How many are there?
There are 602 such terms. - Michael S. Branicky, Dec 31 2021

			Illustration of a(11) = 13751 as a mountain emirp:
  . . . . .
  . . . . .
  . . 7 . .
  . . . . .
  . . . 5 .
  . . . . .
  . 3 . . .
  . . . . .
  1 . . . 1

Michael S. Branicky, Table of n, a(n) for n = 1..602 (full sequence)
G. L. Honaker, Jr. and C. K. Caldwell, Prime Curios! 1231

Cf. A006567, A134941, A134951, A135417, A173071.

# uses A134941()
from sympy import isprime
def is_emirp(n):
    if not isprime(n): return False
    revn = int(str(n)[::-1])
    return n != revn and isprime(revn)
print([k for k in A134941() if is_emirp(k)]) # Michael S. Branicky, Dec 31 2021

A006567 INTERSECT A134941.

More terms from Nathaniel Johnston, Dec 29 2010

Terms a(31) and beyond from Michael S. Branicky, Dec 31 2021

A210548 Larger of emirp pairs whose members have prime digital products.

Original entry on oeis.org

31, 71, 311, 1511, 112111, 1171111, 11131111, 71111111, 131111111, 1115111111, 31111111111, 131111111111111111, 1151111111111111111111111111, 11111111111711111111111111111, 131111111111111111111111111111111111111

Offset: 1

Lekraj Beedassy, Mar 22 2012

Beyond the first two terms, a(n) is the intersection of A173596 and A046703.

Cf. A006567, A210546, A210547.

More terms from Alois P. Heinz, Mar 22 2012

cp's OEIS Frontend

A155513 Lesser of emirps (pairs) with digits 0 and 1 only.

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A156793 Indices of primes whose reversal is a different prime.

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A168159 Distance of the least reversible n-digit prime from 10^(n-1).

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PARI

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A173594 Near-repdigit emirps.

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Mathematica

A175215 The smaller member of a twin prime pair in which both primes are emirps.

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Magma

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A179032 Emirps whose only prime digits are one or more 2's.

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A179033 Emirps with a single 2 as the only prime digit.

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Maple

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A179034 Emirps whose only prime digits are 3's.

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Mathematica

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A182721 Mountain emirps.

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