A109309 -id:A109309 - cp's OEIS frontend

A109308 Lesser emirps (primes whose digit reversal is a larger prime).

13, 17, 37, 79, 107, 113, 149, 157, 167, 179, 199, 337, 347, 359, 389, 709, 739, 769, 1009, 1021, 1031, 1033, 1061, 1069, 1091, 1097, 1103, 1109, 1151, 1153, 1181, 1193, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1279, 1283, 1381, 1399, 1409, 1429

Offset: 1

Zak Seidov, Jun 25 2005

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

Cf. A006567 (emirps), A109309 (larger emirps).

read("transforms"):
A109308 := proc(n)
    option remember;
    local p,R ;
    if n = 1 then
        return 13 ;
    else
        p := nextprime(procname(n-1)) ;
        while true do
            R := digrev(p) ;
            if R> p and isprime(R) then
                return p;
            end if;
            p := nextprime(p) ;
        end do:
    end if;
end proc: # R. J. Mathar, Oct 12 2012

dr[n_]:=FromDigits[Reverse[IntegerDigits[n]]];Select[Prime[Range[1000]], PrimeQ[dr[ # ]]&&dr[ # ]>#&]

isok(p) = if (isprime(p), my(q=fromdigits(Vecrev(digits(p)))); (p < q) && isprime(q)); \\ Michel Marcus, Sep 07 2021

from sympy import isprime, primerange
def ok(p): revp = int(str(p)[::-1]); return p < revp and isprime(revp)
print(list(filter(ok, primerange(1, 1430)))) # Michael S. Branicky, Sep 07 2021

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

A178093 Larger of emirp pairs whose digital sums are also emirps (A178091).

Original entry on oeis.org

733, 751, 953, 971, 1741, 1831, 3271, 3433, 3541, 3613, 3851, 7253, 7321, 7433, 7523, 7901, 9013, 9103, 9161, 9341, 9521, 9769, 9787, 9967, 13711, 14431, 14831, 15241, 15511, 15601, 15731, 16451, 17041, 18701, 19421, 30271, 30491, 30851, 31081, 31481, 31531, 32341

Offset: 1

Lekraj Beedassy, May 19 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A109309, A178091, A178092.

emirpQ[n_] := n != IntegerReverse[n] && PrimeQ[n] && PrimeQ[IntegerReverse[n]]; q[n_] := emirpQ[DigitSum[n]] && n > IntegerReverse[n] && emirpQ[n]; Select[Range[10^5], q] (* Amiram Eldar, Apr 30 2024 *)

More terms from Amiram Eldar, Apr 30 2024

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

A127748 Largest n-digit emirp (A006567) with strictly decreasing (distinct) digits.

Original entry on oeis.org

97, 983, 9871, 98731, 986543, 9875321

Offset: 2

Lekraj Beedassy, Jan 28 2007

Cf. A006567, A030540, A109309, A114019.

Cf. A114019.

Edited and extended by Ray Chandler, Jan 30 2007

A152033 a(n) is the least n-digit prime p whose reversal is a prime q < p.

Original entry on oeis.org

31, 311, 1201, 11701, 106501, 1020101, 10021001, 100440001, 1000200001, 10003810001, 100011400001, 1000039200001, 10000084000001, 100000381000001, 1000001790000001, 10000004700000001, 100000033900000001, 1000000101700000001, 10000000484400000001, 100000001019000000001

Offset: 2

Zak Seidov, Nov 20 2008

Cf. A006567, A152014, A152034.

Subsequence of A109309.

Do[ p = NextPrime[10^(n - 1) ]; Do[ p1 = FromDigits[ Reverse[IntegerDigits[p]]]; If[PrimeQ[p1] && p1 < p, Print[{n, p}]; Break[]]; p = NextPrime[p], {10^9}], {n, 2, 16}];

a(17)-a(21) from Chai Wah Wu, Sep 11 2019

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

cp's OEIS Frontend

A109308 Lesser emirps (primes whose digit reversal is a larger prime).

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

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A178093 Larger of emirp pairs whose digital sums are also emirps (A178091).

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

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A127748 Largest n-digit emirp (A006567) with strictly decreasing (distinct) digits.

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A152033 a(n) is the least n-digit prime p whose reversal is a prime q < p.

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

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