A006567 -id:A006567 - cp's OEIS frontend

A048054 Number of n-digit reversible primes (emirps).

4, 9, 43, 204, 1499, 9538, 71142, 535578, 4197196, 33619380, 274932272, 2294771254, 19489886063, 167630912672, 1456476399463

Offset: 1

Jud McCranie

The count includes palindromes.

			2, 3, 5 and 7 are 1-digit reversible primes, so a(1)=4.

J. L. Boal and J. H. Bevis, Permutable primes. Math. Mag., 55 (N0. 1, 1982), 38-41. [From _N. J. A. Sloane_, Jan 19 2012]
Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 36.

Cf. A003684, A006567, A007628, A046732, A048051, A048052, A048053, A048895.

Count[Range[10^(# - 1), 10^# - 1], n_ /; And[PrimeQ@ n, PrimeQ@ FromDigits@ Reverse@ IntegerDigits@ n]] & /@ Range@ 7 (* Michael De Vlieger, Jul 14 2015 *)

from sympy import isprime, primerange
def A048054(n):
    return len([p for p in primerange(10**(n-1),10**n)
                if isprime(int(str(p)[::-1]))]) # Chai Wah Wu, Aug 14 2014

a(11)-a(13) from Giovanni Resta, Jul 19 2015

a(14)-a(15) from Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Oct 05 2023

A003684 Number of n-digit reversible primes (or emirps) with distinct digits.

Original entry on oeis.org

4, 8, 22, 84, 402, 1218, 3572, 8218, 11804

Offset: 1

Harvey P. Dale and Jud McCranie

			13, 17, 31, 37, 71, 73, 79 and 97 are reversible primes (emirps), so a(2)=8.

Cf. A006567, A007628, A046732, A048051, A048052, A048053, A048054, A048895.

emrpQ[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];rev!=idn && Max[DigitCount[n]] ==1&&PrimeQ[FromDigits[rev]]]; With[{ems=Select[ Prime[ Range[ 51*10^6]],emrpQ]},Join[ {4},Table[Count[ems,?(IntegerLength[ #] == n&)],{n,2,9}]]] (* _Harvey P. Dale, Nov 29 2014 *)

from sympy import primerange, isprime
def A003684(n):
    return len([p for p in primerange(10**(n-1),10**n)
    if len(set(str(p))) == len(str(p)) and isprime(int(str(p)[::-1]))])
# Chai Wah Wu, Aug 14 2014

Typo in example corrected by David Ritterskamp (dritters(AT)usi.edu), Mar 24 2008

A048052 Start of the first occurrence of n consecutive reversible primes (emirps).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 727, 1193, 1193, 1477271183, 9387802769, 15423094826093

Offset: 1

Jud McCranie

Palindromic primes are allowed.

			2, 3, 5, 7, 11, 13 and 17 are consecutive reversible primes, so a(7) = 2.

Carlos Rivera, Problem 17. The largest known sequence of consecutive and reversible primes, Problems and Puzzles.

Cf. A040104 (n=10), A048051 (n=11), A048053 (n=12), A003684, A006567, A007628, A046732, A048054, A048895.

Corrected by Rick L. Shepherd, May 28 2002

a(13) from Giovanni Resta, Nov 07 2019

A048895 Bemirps: primes that yield a different prime when turned upside down with reversals of both being two more different primes.

Original entry on oeis.org

1061, 1091, 1601, 1901, 10061, 10091, 16001, 19001, 106861, 109891, 168601, 198901, 1106881, 1109881, 1606081, 1806061, 1809091, 1886011, 1889011, 1909081, 10806881, 10809881, 11061811, 11091811, 11609681, 11698691, 11816011, 11819011, 11906981

Offset: 1

G. L. Honaker, Jr.

Emirps that yield other emirps when turned upside down. - Lekraj Beedassy, Apr 03 2009
Invertible primes whose reversals are also invertible primes. - Lekraj Beedassy, Apr 04 2009
All terms must begin and end with a one. - T. D. Noe, Apr 21 2014
A term has to include 6 or 9. The concatenation of first n = 809 bemirp 10611091...11688981911 is a prime with 8143 digits being the smallest one for n > 1. There isn't a bemirp < 10^15 with a bemirp index (over all primes). Bemirps such that 4 associated primes are all Sophie Germain primes are 1161880189181, 1191880186181, 1819810881611, 1816810881911, ... . - Metin Sariyar, Mar 06 2020

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000 (first 468 terms from T. D. Noe)
Giovanni Resta, bemirps

Cf. A003684, A006567, A007628, A046732, A048051, A048052, A048053, A048054.

upDown[0] = 0; upDown[1] = 1; upDown[6] = 9; upDown[8] = 8; upDown[9] = 6; fQ[p_] := Module[{revP, upDownP, revUpDownP}, If[Intersection[{2, 3, 4, 5, 7}, Union[IntegerDigits[p]]] != {}, False, revP = FromDigits[Reverse[IntegerDigits[p]]]; upDownP = FromDigits[upDown /@ IntegerDigits[p]]; revUpDownP = FromDigits[Reverse[IntegerDigits[upDownP]]]; p != revP && p != upDownP && p != revUpDownP && PrimeQ[revP] && PrimeQ[upDownP] && PrimeQ[revUpDownP]]]; t = {}; nn = 6; Do[p = 10^n; While[p < 2*10^n, p = NextPrime[p]; If[fQ[p], AppendTo[t, p]]], {n, nn}]; t (* T. D. Noe, Apr 21 2014 *)

More terms from David W. Wilson

A046732 "Norep emirps": primes with distinct digits which remain prime when reversed.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 107, 149, 157, 167, 179, 347, 359, 389, 701, 709, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 1069, 1097, 1237, 1249, 1259, 1279, 1283, 1409, 1429, 1439, 1453, 1487, 1523, 1583, 1597, 1657, 1723, 1753

Offset: 1

Enoch Haga

There are no 10-digit terms because their sum of digits would be 45 and thus the number would be divisible by 3.

There are 25332 terms in this sequence, the last of which is 987653201, as found by Harvey P. Dale. - see Martin Gardner's column in Scientific American.

Nathaniel Johnston, Table of n, a(n) for n = 1..25332 (full sequence).
Chris K. Caldwell and G. L. Honaker, Jr., 987653201, Prime Curios!.
Martin Gardner, Patterns in primes are a clue to the strong law of small numbers, Mathematical Games, Scientific American, Vol. 243, No. 4, September, 1980.
Carlos Rivera, Puzzle 59. Six and the nine digits primes (by Jud McCranie), The Prime Puzzles and Problems Connection.

Intersection of A007500 and A029743.

Cf. A006567, A003684, A007628, A048051, A048052, A048053, A048054, A048895.

read(transforms): A046732 := proc(n) option remember: local d,k,p,distdig: if(n=1)then return 2: fi: p:=procname(n-1): do p:=nextprime(p): if(isprime(digrev(p)))then d:=convert(p,base,10): distdig:=true: for k from 0 to 9 do if(numboccur(d,k)>1)then distdig:=false: break: fi: od: if(distdig)then return p: fi: fi: od: end: seq(A046732(n),n=1..52); # Nathaniel Johnston, May 29 2011

Select[Prime[Range[280]], Length[Union[x = IntegerDigits[#]]] == Length[x] && PrimeQ[FromDigits[Reverse[x]]] &] (* Jayanta Basu, Jun 28 2013 *)

from sympy import prime, isprime
A046732 = [p for p in (prime(n) for n in range(1,10**3)) if len(str(p)) == len(set(str(p))) and isprime(int(str(p)[::-1]))] # Chai Wah Wu, Aug 14 2014

More terms from Jud McCranie.

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

Original entry on oeis.org

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

A007628 Reflectable emirps.

Original entry on oeis.org

13, 31, 113, 311, 1031, 1033, 1103, 1181, 1301, 1381, 1811, 1831, 3011, 3083, 3301, 3803, 10333, 11003, 11083, 11833, 18013, 18133, 18803, 30011, 30881, 31033, 31081, 31183, 33013, 33181, 33301, 33811, 38011, 38113

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Subsequence of A125308, the reflectable primes. - Reinhard Zumkeller, Jul 16 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. W. Trigg, "Reflective Primes", J. Rec. Math., 15 (1983), 251-256.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Prime Glossary, Reflectable prime
C. W. Trigg, Reflectable primes, J. Recreational Mathematics 15.4 (1983), 253. (Annotated scanned copy)

Cf. A003684, A006567, A046732, A048051, A048052, A048053, A048054, A048895.

Cf. A010051, A004086, A125308.

a007628 n = a007628_list !! (n-1)
a007628_list = filter f a125308_list where
   f p = a010051' q == 1 && q /= p  where q = a004086 p
-- Reinhard Zumkeller, Jul 16 2014

Missing 1811 inserted by Reinhard Zumkeller, Jul 16 2014

A109031 Numbers that have exactly eleven prime factors counted with multiplicity (A069272) whose digit reversal is different and also has 11 prime factors (with multiplicity).

Original entry on oeis.org

295245, 426816, 542592, 618624, 2112480, 2116224, 2150064, 2154816, 2196000, 2302560, 2327616, 2342277, 2388672, 2555280, 2576896, 2599200, 2768832, 2952288, 2952576, 4017216, 4074240, 4074840, 4076160, 4076568, 4078848

Offset: 1

Jonathan Vos Post, Jun 16 2005

This sequence is the k = 11 instance of the series which begins with k = 1 (emirps), k = 2, k = 3 (A109023), k = 4 (A109024), k = 5 (A109025), k = 6 (A109026), k = 7 (A109027), k = 8 (A109028), k = 9 (A109029), k = 10 (A109030).

			a(1) = 295245 is in this sequence because 295245 = 3^10 * 5 has exactly 11 prime factors counted with multiplicity and reverse(295245) = 542592 = 2^7 * 3^3 * 157 also has 11 prime factors counted with multiplicity.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

Cf. A069272, A006567, A097393, A109018, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030.

is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 11 && bigomega(r) == 11
} \\ David A. Corneth, Mar 07 2024

a(5)-a(25) from Donovan Johnson, Apr 09 2010

A071612 a(n) is the smallest prime that is the first of n consecutive primes which are all emirps.

Original entry on oeis.org

13, 13, 71, 733, 1193, 1193, 1193, 1193, 1193, 1193, 1477271183, 9387802769, 15423094826093

Offset: 1

Klaus Brockhaus, May 27 2002

			1193,1201,1213,1217,1223,1229,1231,1237,1249,1259 are ten consecutive primes which are all emirps and 1193,1201,1213,1217,1223 is the first occurrence of five consecutive primes which are all emirps, so a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = 1193.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 9387802769

Cf. A006567, A070905, A071614, A103172.

By studying terms of the sequence A103172 we can deduce that a(11), a(12) are 1477271183 & 9387802769. - Farideh Firoozbakht, Jun 17 2010

a(13) from Giovanni Resta, Apr 23 2021

A098562 Primes that are the sum of the squares of the first k primes for some k.

Original entry on oeis.org

13, 20477, 75997, 239087, 2210983, 3579761, 29194283, 40002073, 45448471, 55600481, 77290091, 108095623, 114986483, 155637463, 226226771, 302920139, 324657881, 519681709, 551321299, 618359839, 797005427, 944007487, 1039681147, 1124764853, 1923614047, 2135308631

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

These are the primes arising in A098561.

			From _K. D. Bajpai_, Dec 15 2014: (Start)
13 is in the sequence because the sum of the squares of the first 2 primes is 2^2 + 3^2 = 4 + 9 = 13, which is prime.
20477 is in the sequence because the sum of the squares of the first 18 primes is 2^2 + 3^2 + 5^2 + ... + 59^2 + 61^2 = 4 + 9 + 25 + ... + 3481 + 3721 = 20477, which is prime.
(End)

K. D. Bajpai, Table of n, a(n) for n = 1..15000

Cf. A098561 (corresponding n), A024450 (sum of squares of primes), A066525 (sums of cubes of primes), A013918 (sums of primes).

Cf. A000040, A006567. - Jonathan Vos Post, Aug 13 2009

Select[Table[Sum[Prime[k]^2, {k, 1, n}], {n, 1000}], PrimeQ]  (* K. D. Bajpai, Dec 15 2014 *)

s=0; forprime(p=2, 1e6, t=s+=p^2; if(isprime(t), print1(t,", "))) \\ K. D. Bajpai, Dec 15 2014

a(24)-a(26) from K. D. Bajpai, Dec 15 2014

a(42) in b-file corrected by Andrew Howroyd, Feb 28 2018

cp's OEIS Frontend

A048054 Number of n-digit reversible primes (emirps).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A003684 Number of n-digit reversible primes (or emirps) with distinct digits.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Extensions

A048052 Start of the first occurrence of n consecutive reversible primes (emirps).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A048895 Bemirps: primes that yield a different prime when turned upside down with reversals of both being two more different primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A046732 "Norep emirps": primes with distinct digits which remain prime when reversed.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A007628 Reflectable emirps.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Extensions

A109031 Numbers that have exactly eleven prime factors counted with multiplicity (A069272) whose digit reversal is different and also has 11 prime factors (with multiplicity).

Views

Author

Keywords