A109308 -id:A109308 - cp's OEIS frontend

A343706 Lesser emirps (A109308) p such that A056964(p) is in A343703.

17, 107, 113, 167, 179, 389, 1031, 1091, 1097, 1109, 1181, 1259, 1439, 1487, 1523, 1583, 1619, 1847, 3023, 3089, 3257, 3347, 3359, 3527, 3719, 7349, 7529, 7577, 7589, 7649, 7949, 9029, 10067, 10151, 10247, 10487, 10739, 10781, 11057, 11423, 11621, 11777, 11897, 11933, 12119, 12227, 12641, 13151

Offset: 1

J. M. Bergot and Robert Israel, Apr 26 2021

Primes p such that the digit-reversal q = A004086(p) is a prime greater than p, and p+q = x*y for some x and y such that x+y and the concatenation x|y are primes.

			a(4) = 167 is a term because 167 and 761 are primes with 167 < 761, and 167+761 = 928 = 32*29 with 3229 and 32+29 = 61 prime.

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 1036. P + R(p) such that..., The Prime Puzzles and Problems Connection.

Cf. A004086, A056964, A109308, A343703.

revdigs:= proc(n) local L,i;
    L:= convert(n,base,10);
    add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(p) local q,m,d,e;
   q:= revdigs(p); if q <= p then return false fi;
   if not isprime(p) or not isprime(q) then return false fi;
   m:= p+q;
   for d in numtheory:-divisors(m) do
     e:= m/d;
     if isprime(d*10^(1+ilog10(e))+e) and isprime(d+e) then return true fi
   od;
   false
end proc:

A346756 Lesser emirps (A109308) subtracted from their reversals.

Original entry on oeis.org

18, 54, 36, 18, 594, 198, 792, 594, 594, 792, 792, 396, 396, 594, 594, 198, 198, 198, 7992, 180, 270, 2268, 540, 8532, 810, 6804, 1908, 7902, 360, 2358, 630, 2718, 1908, 5904, 1998, 7992, 90, 6084, 8172, 8262, 8442, 2538, 450, 8532, 7632, 7812, 7902, 2088, 270

Offset: 1

George Bull, Aug 20 2021

			31 - 13 = 18, 71 - 17 = 54, 73 - 37 = 36 (distance between lesser emirps and their reversals).

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 20. Reversible Primes, The Prime Puzzles and Problems Connection.

Cf. A004086, A006567, A109308.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L));
end proc:
f:= proc(p) local r;
  if not isprime(p) then return NULL fi;
  r:= rev(p);
  if r > p and isprime(r) then r-p else NULL fi
end proc:
map(f, [seq(i,i=11 .. 10^4, 2)]); # Robert Israel, Dec 28 2023

f[n_] := IntegerReverse[n] - n; Map[f, Select[Range[1500], f[#] > 0 && PrimeQ[#] && PrimeQ @ IntegerReverse[#] &]] (* Amiram Eldar, Sep 08 2021 *)

rev(p) = fromdigits(Vecrev(digits(p))); \\ A004086
lista(nn) = {my(list = List()); forprime (p=1, nn, my(q=rev(p)); if ((q>p) && isprime(q), listput(list, q-p));); Vec(list);} \\ Michel Marcus, Sep 07 2021

from sympy import isprime, nextprime
def aupton(terms):
    alst, p = [], 2
    while len(alst) < terms:
        revp = int(str(p)[::-1])
        if p < revp and isprime(revp):
            alst.append(revp - p)
        p = nextprime(p)
    return alst
print(aupton(49)) # Michael S. Branicky, Sep 08 2021

a(n) = reverse(A109308(n)) - A109308(n).

Better name and more terms from Jon E. Schoenfield, Aug 20 2021

A083815 Semiprimes whose prime factors are distinct and the reversal of one factor is equal to the other.

Original entry on oeis.org

403, 1207, 2701, 7663, 35143, 75007, 117907, 127087, 140209, 173809, 197209, 247021, 257821, 342127, 382387, 643063, 692443, 743623, 1226221, 1341331, 1626151, 1698661, 1739161, 2073991, 2138791, 2528611, 2561011, 3321133

Offset: 1

Jason Earls, Jun 17 2003

Products of emirp pairs, sorted. - Lekraj Beedassy, Jan 10 2008

			a(2)= 1207 = 17 * 71.

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

Cf. A001358, A006567, A109308.

revdigs:= proc(n) local i,L;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
f:= proc(p) local r;
  if not isprime(p) then return NULL fi;
  r:= revdigs(p);
  if r > p and isprime(r) then r*p fi
end proc:
sort(map(f, [seq(i,i=13..9999,2)])); # Robert Israel, Dec 26 2018

More terms from Ray Chandler, Jul 22 2003

A109309 Larger emirps (primes whose digit reversal is a lesser prime).

Original entry on oeis.org

31, 71, 73, 97, 311, 701, 733, 743, 751, 761, 907, 937, 941, 953, 967, 971, 983, 991, 1201, 1301, 1321, 1511, 1601, 1741, 1811, 1831, 1901, 3011, 3121, 3191, 3203, 3221, 3251, 3271, 3301, 3371, 3391, 3433, 3511, 3541, 3571, 3613, 3643, 3733, 3803, 3821

Offset: 1

Zak Seidov, Jun 25 2005

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

Cf. A006567 Emirps (primes whose digit reversal is a different prime), A109308 Lesser emirps (primes whose digit reversal is a larger prime).

dr[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Select[Prime[Range[1000]], PrimeQ[dr[ # ]]&&dr[ # ]<#&]
Select[Prime[Range[600]],PrimeQ[IntegerReverse[#]]&&#>IntegerReverse[#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 23 2020 *)

A178581 Primes that are the average of the members of emirp pairs.

Original entry on oeis.org

11311, 12721, 13831, 14741, 16061, 16561, 17471, 18481, 20507, 21107, 21407, 21617, 21817, 22727, 23027, 23227, 23327, 23537, 24137, 24547, 24847, 25147, 25247, 25447, 25657, 26357, 27067, 27367, 28277, 34543, 34843, 35153, 35353

Offset: 1

Lekraj Beedassy, May 29 2010

For the lesser member of the corresponding smallest emirp pair, see A178582.

Cf. A006567, A109308, A109309

A178092 Lesser of emirp pairs whose digital sums are also emirps (A178091).

Original entry on oeis.org

157, 179, 337, 359, 1097, 1237, 1259, 1381, 1439, 1453, 1471, 1583, 1619, 1723, 3019, 3109, 3163, 3257, 3343, 3347, 3527, 7699, 7879, 9679, 10039, 10079, 10273, 10453, 10457, 10651, 10781, 10853, 11083, 11159, 11353, 11447, 11551, 11717, 11731, 11933, 12073

Offset: 1

Lekraj Beedassy, May 19 2010

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A109308, A178091, A178093.

emrp[n_]:=Module[{e=FromDigits[Reverse[IntegerDigits[n]]]},If[n!=e&&PrimeQ[ e],{n,e},{}]]; dsemQ[m_]:=Module[ {ds=Total[IntegerDigits[m]],em}, em= FromDigits[ Reverse[IntegerDigits[ ds]]];em!=ds&& And@@PrimeQ[{ds,em}]]; Select[Transpose[Rest[Union[Sort/@(emrp/@Prime[Range[2000]])]]][[1]],dsemQ] (* Harvey P. Dale, Sep 11 2013 *)

Corrected and extended by Harvey P. Dale, Sep 11 2013

A166681 Primes p which have at least one prime anagram larger than p.

Original entry on oeis.org

13, 17, 37, 79, 107, 113, 127, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 191, 197, 199, 239, 241, 251, 277, 281, 283, 313, 337, 347, 349, 359, 367, 373, 379, 389, 397, 419, 457, 461, 463, 467, 479, 491, 563, 569, 571, 577, 587, 593, 613

Offset: 1

Pierre CAMI, Oct 18 2009

Primes like 113, 137, 149, 157 etc have more than one such larger anagram, but are only listed once.

			13 is the first with 31 as prime anagram.
17 is the second with 71 as prime anagram.
31 has one anagram 13 but this is not >31 so 31 is not in the sequence.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A055387, A109308, A069567.

filter:= proc(p) local L,Lp,q,i;
  if not isprime(p) then return false fi;
  L:= convert(p,base,10);
  for Lp in combinat:-permute(L) do
    q:= add(Lp[i]*10^(i-1),i=1..nops(L));
    if q > p and isprime(q) then return true fi
  od;
false
end proc:
select(filter, [seq(i,i=13..1000,2)]); # Robert Israel, Jan 18 2023

paQ[n_]:=Length[Select[FromDigits/@Permutations[IntegerDigits[n]],#>n && PrimeQ[#]&]]>0; Select[Prime[Range[200]],paQ] (* Harvey P. Dale, Sep 23 2013 *)

from itertools import islice
from sympy.utilities.iterables import multiset_permutations
from sympy import isprime, nextprime
def A166681_gen(): # generator of terms
    p = 13
    while True:
        for q in multiset_permutations(str(p)):
            if (r:=int(''.join(q)))>p and isprime(r):
                yield p
                break
        p = nextprime(p)
A166681_list = list(islice(A166681_gen(),20)) # Chai Wah Wu, Jan 17 2023

Definition clarified, sequence extended. - R. J. Mathar, Oct 12 2012

A127747 Smallest n-digit emirp (A006567) with strictly increasing (distinct) digits.

Original entry on oeis.org

13, 149, 1237, 12689, 345689, 1235789

Offset: 2

Lekraj Beedassy, Jan 28 2007

Cf. A006567, A031991, A109308, A114018, A127746.

Edited and extended by Ray Chandler, Jan 30 2007

A246944 Prime numbers p with property that when the first digit is shifted to the last digit, we obtain a prime greater than p.

Original entry on oeis.org

13, 17, 37, 79, 113, 127, 131, 149, 157, 163, 181, 191, 197, 199, 337, 359, 367, 373, 787, 797, 1117, 1123, 1129, 1151, 1153, 1181, 1187, 1193, 1201, 1213, 1231, 1237, 1259, 1279, 1297, 1301, 1319, 1327, 1367, 1409, 1423, 1427, 1439, 1459, 1483, 1487, 1493

Offset: 1

Shanmuga Subramanian, Sep 08 2014

			13: when 1 is shifted to last it becomes 31.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A109308.

fdlQ[n_]:=Module[{c=FromDigits[RotateLeft[IntegerDigits[n]]]},PrimeQ[c] && c>n]; Select[Prime[Range[300]],fdlQ] (* Harvey P. Dale, Jun 06 2018 *)

move(n)=n=Vec(Str(n));N=List();for(i=2,#n,listput(N,n[i]));listput(N,n[1]);return(eval(concat(N)));
forprime(p=2,1000,if(isprime(move(p))&&move(p)>p,print1(p,","))) \\ Edward Jiang, Sep 08 2014

function number_firsttolast($a)
{
$b=str_split($a);
$c=count($b);
for($i=1;$i<$c;$i++)
{
  $d=$d.$b[$i];
}
$e=$d.$b[0];
return ($e);
}

A263241 Lesser of emirp pairs that are merely reversals of their end digits.

Original entry on oeis.org

13, 17, 37, 79, 107, 113, 149, 157, 167, 179, 199, 337, 347, 359, 389, 709, 739, 769, 1009, 1223, 1229, 1559, 1669, 3889, 7229, 10007, 10009, 10909, 11717, 11719, 12227, 12323, 12829, 13337, 13933, 14143, 14447, 14449, 14549, 14843, 14947, 15053, 16063, 16267, 16567, 16763

Offset: 1

Lekraj Beedassy, Oct 13 2015

The first digit is always smaller than last digit.

Cf. A006567, A109308, A263240, A263242.

Select[Range[20000], And @@ PrimeQ[{#, IntegerReverse[#]}] && ! PalindromeQ[#] && First[(d = IntegerDigits[#])] < Last[d] && PalindromeQ[Most@ Rest@ IntegerDigits[#]] &] (* Amiram Eldar, Oct 16 2021 *)

isok(p) = my(dp=digits(p), dr=Vecrev(dp), r=fromdigits(dr)); if (isprime(r) && (r>p) && isprime(p), sum(i=2, #dp-1, dp[i]==dr[i]) == #dp-2); \\ Michel Marcus, Oct 16 2021

Missing 179 added by Zak Seidov, Oct 15 2021

cp's OEIS Frontend

A343706 Lesser emirps (A109308) p such that A056964(p) is in A343703.

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A346756 Lesser emirps (A109308) subtracted from their reversals.

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A083815 Semiprimes whose prime factors are distinct and the reversal of one factor is equal to the other.

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Maple

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A109309 Larger emirps (primes whose digit reversal is a lesser prime).

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Mathematica

A178581 Primes that are the average of the members of emirp pairs.

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A178092 Lesser of emirp pairs whose digital sums are also emirps (A178091).

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Mathematica

Extensions

A166681 Primes p which have at least one prime anagram larger than p.

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Programs

Maple

Mathematica

Python

Extensions

A127747 Smallest n-digit emirp (A006567) with strictly increasing (distinct) digits.

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Author

Keywords

Crossrefs

Extensions

A246944 Prime numbers p with property that when the first digit is shifted to the last digit, we obtain a prime greater than p.

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