A006567 -id:A006567 - cp's OEIS frontend

A155509 Larger of emirps (pairs) with digits 1 and 9 only.

991, 99119, 191911, 911111, 919111, 999199, 1911911, 1991911, 9111911, 11911111, 19911191, 19919111, 19991911, 99111119, 911111191, 991111111, 991991111, 999199991, 999919919, 1911119911, 1919991191, 1991111911, 1991919191

Offset: 1

Lekraj Beedassy, Jan 23 2009

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

A006567, A155507, A155508

le19[n_]:=Module[{emrps=Select[FromDigits/@Tuples[{1,9},n],!PalindromeQ[ #] && AllTrue[ {#,IntegerReverse[#]},PrimeQ]&]},If[IntegerReverse[#]>#, IntegerReverse[ #],{}]&/@emrps/.{}->Nothing]; Flatten[Table[le19[x],{x,10}]] // Sort (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 04 2019 *)

More terms from Sean A. Irvine, Apr 06 2010

A155514 Larger of emirps (pairs) with digits 0 and 1 only.

Original entry on oeis.org

10111001, 111001001, 1010110001, 1011000101, 1101001001, 1111100101, 10100101001, 11000000101, 11010011101, 11100000101, 100111101001, 101010000001, 101010111001, 110000000101, 110011010101, 110101011001

Offset: 1

Lekraj Beedassy, Jan 23 2009

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

A006567, A155512, A155513

emrpQ[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn]; idn!=ridn &&PrimeQ[n]&&PrimeQ[FromDigits[ridn]]]; lrgr[n_]:=If[nHarvey P. Dale, Oct 01 2012 *)

First missed entry added. Lekraj Beedassy, May 30 2009

More terms from Sean A. Irvine, Mar 04 2010

A178091 Emirps whose digital sums are also emirps.

Original entry on oeis.org

157, 179, 337, 359, 733, 751, 953, 971, 1097, 1237, 1259, 1381, 1439, 1453, 1471, 1583, 1619, 1723, 1741, 1831, 3019, 3109, 3163, 3257, 3271, 3343, 3347, 3433, 3527, 3541, 3613, 3851, 7253, 7321, 7433, 7523, 7699, 7879, 7901, 9013, 9103, 9161, 9341, 9521, 9679, 9769, 9787, 9967

Offset: 1

Lekraj Beedassy, May 19 2010

Palindromic primes are not allowed, nor are palindromic digital sums of primes. - Harvey P. Dale, Feb 23 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Silva, Prime Curios, Entry 157.

Cf. A006567, A178092, A178093.

dseQ[n_]:=Module[{idn=IntegerDigits[n],ds},ds=IntegerDigits[Total[idn]];idn!=Reverse[idn]&&ds!=Reverse[ds] && And@@PrimeQ[{FromDigits[ Reverse[ idn]],FromDigits[ds],FromDigits[Reverse[ds]]}]]; Select[Prime[Range[ 1300]],dseQ] (* Harvey P. Dale, Feb 23 2014 *)

A185439 Emirp gaps: Differences between consecutive emirps.

Original entry on oeis.org

4, 14, 6, 34, 2, 6, 18, 10, 6, 36, 8, 10, 12, 20, 112, 26, 10, 12, 30, 312, 8, 24, 6, 4, 8, 10, 8, 138, 30, 4, 12, 14, 4, 12, 8, 18, 12, 10, 2, 28, 8, 22, 6, 6, 6, 42, 2, 28, 12, 8, 12, 4, 6, 6, 2, 6, 12, 10, 20, 4, 18, 20, 60, 18, 10, 20, 10, 14, 18, 16, 12, 12, 12, 36, 24, 14, 4, 18, 38, 12, 54, 10, 8, 12, 36, 22, 20

Offset: 1

Jonathan Vos Post, Feb 03 2011

Gaps between consecutive primes whose reversal is a different prime. This is to Differences between consecutive primes (A001223) as emirps (A006567) are to primes (A000040). This was indirectly suggested to me in a facebook conversation with Kevin L. Schwartz. One may use this to derive other sequences: records in emirp gaps; lower of pair of consecutive emirps with record gap; larger of pair of emirps with record gaps, by analogy with A005250, A002386, A000101.

			The first 9 emirps are 13, 17, 31, 37, 71, 73, 79, 97, 107.
Hence the first 8 gaps between consecutive emirps are:
   17 - 13 =  4;
   31 - 17 = 14;
   37 - 31 =  6;
   71 - 37 = 34;
   73 - 71 =  2 (i.e., 71 and 73 are a pair of "twin prime emirps");
   79 - 73 =  6;
   97 - 79 = 18;
  107 - 97 = 10.
So far, we see a minimum gap of 2, and a maximum of 34.

Metin Sariyar, Table of n, a(n) for n = 1..10000

Cf. A000040, A000101, A001223, A002386, A006567.

emirpQ[n_]:=Module[{idn=IntegerDigits[n],ridn},ridn=Reverse[idn];idn!=ridn&&PrimeQ[FromDigits[ridn]]]
Take[Differences[Select[Prime[Range[1000]],emirpQ]],90]  (* Harvey P. Dale, Feb 18 2011 *)

a(n) = A006567(n+1) - A006567(n).

A210546 Emirps whose products of digits are prime.

Original entry on oeis.org

13, 17, 31, 71, 113, 311, 1151, 1511, 111211, 112111, 1111711, 1171111, 11111117, 11113111, 11131111, 71111111, 111111131, 131111111, 1111115111, 1115111111, 11111111113, 31111111111, 111111111111111131, 131111111111111111, 1111111111111111111111111511

Offset: 1

Lekraj Beedassy, Mar 22 2012

Chai Wah Wu, Table of n, a(n) for n = 1..66

Cf. A006567, A046703, A210547, A210548.

from _future_ import division
from sympy import isprime
A210546_list = []
for l in range(1,20):
    q = (10**l-1)//9
    for i in range(l):
        for p in [2,3,5,7]:
            r = q+(p-1)*10**i
            s, t = str(r), str(r)[::-1]
            if s != t and isprime(r) and isprime(int(t)):
                A210546_list.append(r) # Chai Wah Wu, Aug 15 2017

5 more terms from Alois P. Heinz, Mar 29 2012

A210547 Lesser of emirp pairs whose members have prime digital products.

Original entry on oeis.org

13, 17, 113, 1151, 111211, 1111711, 11111117, 11113111, 111111131, 1111115111, 11111111113, 111111111111111131, 1111111111111111111111111511, 11111111111111111711111111111, 111111111111111111111111111111111111131

Offset: 1

Lekraj Beedassy, Mar 22 2012

Beyond the first two terms, a(n) is the intersection of A173595 and A046703.

Cf. A006567, A210546, A210548.

More terms from Alois P. Heinz, Mar 22 2012

A263242 Larger of emirp pairs that are merely reversals of their end digits.

Original entry on oeis.org

31, 71, 73, 97, 311, 701, 733, 743, 751, 761, 907, 937, 941, 953, 967, 971, 983, 991, 3221, 9001, 9221, 9227, 9551, 9661, 9883, 32321, 33931, 34141, 34841, 35051, 36061, 36761, 37571, 39791, 70001, 71711, 72221, 73331, 74143, 74441, 74843, 74941, 75253, 76261, 76463, 76561

Offset: 1

Lekraj Beedassy, Oct 13 2015

The first digit is always larger than the last digit.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A006567, A109309, A263240, A263241.

epQ[n_]:=Module[{idn=IntegerDigits[n],mid},mid=Rest[Most[idn]];PrimeQ[ IntegerReverse[n]]&&mid==Reverse[mid]&&idn[[1]]>idn[[-1]]]; Select[ Prime[Range[6,8000]],epQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 10 2016 *)

Corrected by Harvey P. Dale, Oct 10 2016

A345408 Numbers that are the sum of an emirp and its reversal in more than one way.

Original entry on oeis.org

1090, 2662, 2992, 3212, 4334, 4994, 5104, 5324, 6776, 7106, 9328, 9548, 10450, 10670, 10780, 11110, 11330, 11440, 11660, 12122, 12452, 12892, 13222, 15004, 16786, 17446, 17666, 29092, 29482, 31912, 36352, 44644, 44834, 45454, 46654, 46664, 47474, 47864, 49094, 49294, 49484, 49684, 49894, 50104

Offset: 1

J. M. Bergot and Robert Israel, Jun 18 2021

Numbers that are in A345409 in more than one way.

Interchanging an emirp and its reversal is not counted as a different way.

			a(3) = 2992 is a member because 2992 = 1091 + 1901 = 1181+1811 where 1091 and 1181 and their reversals 1901 and 1811 are primes.

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

Cf. A006567, A345409.

revdigs:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc:
isemirp1:= proc(n) local r;
if not isprime(n) then return false fi;
r:= revdigs(n);
r > n and isprime(r)
end proc:
E:= select(isemirp1, [seq(seq(seq(i*10^d+j,j=1..10^d-1,2),i=[1,3,7,9]),d=1..4)]):
V:= sort(map(t -> t+revdigs(t),E)):
M:= select(t -> V[t+1]=V[t], [$1..nops(V)-1]):
sort(convert(convert(V[M],set),list));

from collections import Counter
from sympy import isprime, nextprime
def epgen(start=1, end=float('inf')): # generates unique emirp/prime pairs
    p = nextprime(start-1)
    while p <= end:
        revp = int(str(p)[::-1])
        if p < revp and isprime(revp): yield (p, revp)
        p = nextprime(p)
def aupto(lim):
    c = Counter(sum(ep) for ep in epgen(1, lim) if sum(ep) <= lim)
    return sorted(s for s in c if c[s] > 1)
print(aupto(50105)) # Michael S. Branicky, Jun 18 2021

A345409 Numbers that are the sum of an emirp and its reversal.

Original entry on oeis.org

44, 88, 110, 176, 424, 808, 908, 928, 1070, 1090, 1150, 1190, 1312, 1372, 1616, 1676, 1736, 2222, 2332, 2552, 2662, 2992, 3212, 4114, 4334, 4444, 4664, 4774, 4994, 5104, 5324, 5434, 6226, 6776, 6886, 7106, 7436, 8338, 8558, 8998, 9218, 9328, 9548, 10010, 10120, 10450, 10670, 10780, 11000, 11110

Offset: 1

J. M. Bergot and Robert Israel, Jun 18 2021

nonn

			a(3) = 110 is a member because 110 = 37+73 where 37 is an emirp.

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

Cf. A006567, A345408.

revdigs:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc:
f:= proc(n) local r;
if not isprime(n) then return NULL fi;
r:= revdigs(n);
if r > n and isprime(r) then return r+n fi
end proc:
S:= map(f, {seq(seq(seq(i*10^d+j,j=1..10^d-1,2),i=[1,3,7,9]),d=1..4)}):
sort(convert(S,list));

from sympy import isprime, nextprime
def epgen(start=1, end=float('inf')): # generates unique emirp/prime pairs
    p = nextprime(start-1)
    while p <= end:
        revp = int(str(p)[::-1])
        if p < revp and isprime(revp): yield (p, revp)
        p = nextprime(p)
def aupto(lim):
    epsums = set(sum(ep) for ep in epgen(1, lim))
    return sorted(filter(lambda x: x<=lim, epsums))
print(aupto(11111)) # Michael S. Branicky, Jun 18 2021

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

cp's OEIS Frontend

A155509 Larger of emirps (pairs) with digits 1 and 9 only.

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A155514 Larger of emirps (pairs) with digits 0 and 1 only.

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Mathematica

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A178091 Emirps whose digital sums are also emirps.

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Mathematica

A185439 Emirp gaps: Differences between consecutive emirps.

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Mathematica

Formula

A210546 Emirps whose products of digits are prime.

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Python

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A210547 Lesser of emirp pairs whose members have prime digital products.

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A263242 Larger of emirp pairs that are merely reversals of their end digits.

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Mathematica

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A345408 Numbers that are the sum of an emirp and its reversal in more than one way.

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Maple

Python

A345409 Numbers that are the sum of an emirp and its reversal.

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