A002385 -id:A002385 - cp's OEIS frontend

A007500 Primes whose reversal in base 10 is also prime (called "palindromic primes" by David Wells, although that name usually refers to A002385). Also called reversible primes.

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 101, 107, 113, 131, 149, 151, 157, 167, 179, 181, 191, 199, 311, 313, 337, 347, 353, 359, 373, 383, 389, 701, 709, 727, 733, 739, 743, 751, 757, 761, 769, 787, 797, 907, 919, 929, 937, 941, 953, 967, 971, 983, 991, 1009, 1021

Offset: 1

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

The numbers themselves need not be palindromes.
The range is a subset of the range of A071786. - Reinhard Zumkeller, Jul 06 2009
Number of terms less than 10^n: 4, 13, 56, 260, 1759, 11297, 82439, 618017, 4815213, 38434593, ..., . - Robert G. Wilson v, Jan 08 2015

Roozbeh Hazrat, Mathematica: A Problem-Centered Approach, Springer 2010, pp. 39, 131-132
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 134.

T. D. Noe, Table of n, a(n) for n = 1..10000
Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, and Cathy Swaenepoel, Reversible primes, arXiv:2309.11380 [math.NT], 2023. See p. 3.

Cf. A006567, A007628.
Cf. A002385 (primes that are palindromes in base 10).
Equals A002385 union A006567.
Complement of A076056 with respect to A000040. [From Reinhard Zumkeller, Jul 06 2009]
Cf. A004086, A010051, A000040.

a007500 n = a007500_list !! (n-1)
a007500_list = filter ((== 1) . a010051 . a004086) a000040_list
-- Reinhard Zumkeller, Oct 14 2011

[ p: p in PrimesUpTo(1030) | IsPrime(Seqint(Reverse(Intseq(p)))) ];  // Bruno Berselli, Jul 08 2011

revdigs:= proc(n)
local L,nL,i;
L:= convert(n,base,10);
nL:= nops(L);
add(L[i]*10^(nL-i),i=1..nL);
end:
Primes:= select(isprime,{2,seq(2*i+1,i=1..5*10^5)}):
Primes intersect map(revdigs,Primes); # Robert Israel, Aug 14 2014

Select[ Prime[ Range[ 168 ] ], PrimeQ[ FromDigits[ Reverse[ IntegerDigits[ # ] ] ] ]& ] (* Zak Seidov, corrected by T. D. Noe *)
Select[Prime[Range[1000]],PrimeQ[IntegerReverse[#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 15 2016 *)

is_A007500(n)={ isprime(n) & is_A095179(n)} \\ M. F. Hasler, Jan 13 2012

from sympy import prime, isprime
A007500 = [prime(n) for n in range(1,10**6) if isprime(int(str(prime(n))[::-1]))] # Chai Wah Wu, Aug 14 2014

from gmpy2 import is_prime, mpz
from itertools import count, islice, product
def agen(): # generator of terms
    yield from [2, 3, 5, 7]
    p = 11
    for digits in count(2):
        for first in "1379":
            for mid in product("0123456789", repeat=digits-2):
                for last in "1379":
                    s = first + "".join(mid) + last
                    if is_prime(t:=mpz(s)) and is_prime(mpz(s[::-1])):
                        yield int(t)
print(list(islice(agen(), 60))) # Michael S. Branicky, Jan 02 2025

More terms from Larry Reeves (larryr(AT)acm.org), Oct 31 2000
Added further terms to the sequence Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 16 2009. Checked by N. J. A. Sloane, Jan 20 2009.
Third reference added by Harvey P. Dale, Oct 17 2011

A192139 Powers p^m, m >= 0, of palindromic primes p (A002385).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 25, 27, 32, 49, 64, 81, 101, 121, 125, 128, 131, 151, 181, 191, 243, 256, 313, 343, 353, 373, 383, 512, 625, 727, 729, 757, 787, 797, 919, 929, 1024, 1331, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 10201, 10301, 10501, 10601, 11311, 11411, 12421, 12721

Offset: 1

Jaroslav Krizek, Jun 24 2011

Superset of A002385 and A084092. Subset of A192137.

M. F. Hasler, Table of n, a(n) for n = 1..858 (all terms up to 10^7).

Cf. A002113, A002385, A084317, A084092, A192137, A192138, A192140, A192141.

A192139=[1]; forprime(p=1,M=1e7, p==A004086(p) && A192139=setunion(A192139,vector(log(M)\log(p),k,p^k))) \\ M. F. Hasler, May 11 2015

Missing term 625 inserted and more terms added by M. F. Hasler, May 11 2015

A046405 Numbers of the form pqr where p,q,r are distinct odd palindromic primes (odd terms from A002385).

Original entry on oeis.org

105, 165, 231, 385, 1515, 1965, 2121, 2265, 2715, 2751, 2865, 3171, 3333, 3535, 3801, 4011, 4323, 4585, 4695, 4983, 5285, 5295, 5555, 5595, 5745, 5973, 6303, 6335, 6573, 6685, 7205, 7413, 7777, 7833, 8043, 8305, 9955, 10087, 10329, 10505, 10905

Offset: 1

Patrick De Geest, Jun 15 1998

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Intersection of A033620 and A046389.

Cf. A002385, A046373.

Definition clarified by N. J. A. Sloane, Dec 19 2017 at the suggestion of Harvey P. Dale

Offset changed by Andrew Howroyd, Aug 14 2024

A046485 Sum of first n palindromic primes A002385.

Original entry on oeis.org

2, 5, 10, 17, 28, 129, 260, 411, 592, 783, 1096, 1449, 1822, 2205, 2932, 3689, 4476, 5273, 6192, 7121, 17422, 27923, 38524, 49835, 61246, 73667, 86388, 99209, 112540, 126371, 140302, 154643, 169384, 184835, 200386, 216447, 232808, 249369, 266030, 283501

Offset: 1

Patrick De Geest, Sep 15 1998

The subsequence of prime partial sum of palindromic primes begins: 2, 5, 17, 5273, 7121, 154643, 283501. What is the smallest nontrivial (i.e., multidigit) palindromic prime partial sum of palindromic primes? [Jonathan Vos Post, Feb 07 2010]

Vincenzo Librandi, Table of n, a(n) for n = 1..700
Patrick De Geest, World!Of Palindromic Primes

Cf. A002385, A007504, A046489.

Cf. A000040, A007500, A006567, A016041, A029732, A117697. [Jonathan Vos Post, Feb 07 2010]

t = {}; b = 10; Do[p = Prime[n]; i = IntegerDigits[p, b]; If[i == Reverse[i], AppendTo[t, p];(*Print[p.FromDigits[i]]*)], {n, 4000}]; Accumulate[t] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
Accumulate[Select[Prime[Range[10000]],IntegerDigits[#]==Reverse[ IntegerDigits[#]]&]] (* Harvey P. Dale, Aug 10 2013 *)

a(n) = Sum_{i=1..n} A002385(i) = Sum_{i=1..n} {p prime and R(p) = p, i.e., primes whose decimal expansion is a palindrome}. [Jonathan Vos Post, Feb 07 2010]

Offset set to 1 by R. J. Mathar, Feb 21 2010

A046373 Numbers of the form pqr where p,q,r are (not necessarily distinct) odd palindromic primes (odd terms from A002385).

Original entry on oeis.org

27, 45, 63, 75, 99, 105, 125, 147, 165, 175, 231, 245, 275, 343, 363, 385, 539, 605, 847, 909, 1179, 1331, 1359, 1515, 1629, 1719, 1965, 2121, 2265, 2525, 2715, 2751, 2817, 2865, 3171, 3177, 3275, 3333, 3357, 3447, 3535, 3775, 3801, 4011, 4323, 4525

Offset: 1

Patrick De Geest, Jun 15 1998

Cf. A002385, A046316, A046405.

Take[Times@@@Tuples[Select[Prime[Range[2,100]],PalindromeQ],3]//Union,50] (* Harvey P. Dale, Sep 10 2019 *)

Definition clarified by N. J. A. Sloane, Dec 19 2017 at the suggestion of Harvey P. Dale

A118064 Decimal expansion of the sum of the reciprocals of the palindromic primes A002385 (Honaker's constant).

Original entry on oeis.org

1, 3, 2, 3, 9, 8, 2, 1, 4, 6, 8, 0, 6

Offset: 1

Martin Renner, May 11 2006

From Robert G. Wilson v, Nov 01 2010: (Start)
n \ sum to 10^n
1.267099567099567099567099567099567099567099567099567099567099567099567
1.320723244590290964212793334437872849720871258315369002493912638038324
1.323748402250648554164425746280035962754669829327727800040192015109270
1.323964105671202458016249150576217276147952428601889817773483085610332
1.323980718065525060936354534562000413901564393192688451911141729415146
1.323982026479475203850120990923294207966175748395470136325039323549015
1.323982136437462724794656629740867909978221153827990721566573347887836
1.323982145891606234777299440047139038371441916546100653011463101470839
1.323982146724859090645464845257681674740147563533254654075059843860490
1.323982146799188851138232927173756400348958236915409881890097448921521
1.323982146805857558347279363344557427339916178257233985191868031567947 (End)

			1.323982146806...

Carlos Rivera, Problems & Puzzles: Puzzle 056 - Honaker's Constant.
Eric Weisstein, Palindromic Prime.

Cf. A002385, A160910, A181442, A050251, A118031, A194097.

(* first obtain nextPalindrome from A007632 *) s = 1/11; c = 1; pp = 1; Do[ While[pp < 10^n, If[PrimeQ@ pp, c++; s = N[s + 1/pp, 64]]; pp = NextPalindrome@ pp]; If[ OddQ@ n, pp = 10^(n + 1); Print[{s, n, c}]], {n, 17}] (* Robert G. Wilson v, May 31 2009 *)
generate[n_] := Block[{id = IntegerDigits@n, insert = {{0}, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}}}, FromDigits@ Join[id, #, Reverse@ id] & /@ insert]; sm = N[Plus @@ (1/{2, 3, 5, 7, 11}), 64]; k = 1; Do [While[k < 10^n, sm = N[sm + Plus @@ (1/Select[ generate@k, PrimeQ]), 128]; k++ ]; Print[{2 n + 1, sm}], {n, 9}] (* Robert G. Wilson v, Nov 01 2010 *)

Equals Sum_{p} 1/p, where p ranges over the palindromic primes.

Corrected by Eric W. Weisstein, May 14 2006
More terms from Robert G. Wilson v, Nov 01 2010
Entry revised by N. J. A. Sloane, May 05 2013

A287961 Numbers that are the sum of two palindromic primes (A002385).

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 22, 103, 104, 106, 108, 112, 133, 134, 136, 138, 142, 153, 154, 156, 158, 162, 183, 184, 186, 188, 192, 193, 194, 196, 198, 202, 232, 252, 262, 282, 292, 302, 312, 315, 316, 318, 320, 322, 324, 332, 342, 355, 356, 358, 360, 362, 364, 372, 375, 376, 378, 380

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

Eric Weisstein's World of Mathematics, Palindromic Prime
Index entries for sequences related to palindromes

Cf. A002113, A002385, A014091, A035137, A260254, A260255, A261906.

nmax = 380; f[x_] := Sum[Boole[PalindromeQ[k] && PrimeQ[k]] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A327428 Record gaps between palindromic primes (lower end): primes A002385(k) where A002385(k+1) - A002385(k) exceeds A002385(j+1) - A002385(j) for all j < k.

Original entry on oeis.org

2, 3, 5, 11, 191, 383, 929, 19991, 39293, 98689, 1998991, 3998993, 9989899, 199999991, 399878993, 999727999, 19998289991, 39998389993, 99999199999, 1999987899991, 3999992999993, 9999987899999, 199999959999991, 399999959999993, 999999787999999, 19999998589999991, 39999999299999993

Offset: 1

Chai Wah Wu, Sep 09 2019

This sequence is infinite if there exists an infinite number of palindromic primes (see comment in A327427).

Cf. A002385, A002386, A327427.

A286970 Number of compositions (ordered partitions) of n into decimal palindromic primes (A002385).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 6, 6, 10, 16, 20, 35, 45, 72, 103, 150, 228, 324, 491, 710, 1053, 1552, 2272, 3369, 4930, 7288, 10711, 15771, 23244, 34175, 50382, 74113, 109168, 160722, 236596, 348446, 512894, 755303, 1111946, 1637205, 2410592, 3549023, 5225659, 7693623, 11327912

Offset: 0

Ilya Gutkovskiy, May 17 2017

			a(7) = 6 because we have [7], [5, 2], [3, 2, 2], [2, 5], [2, 3, 2] and [2, 2, 3].

Eric Weisstein's World of Mathematics, Palindromic Prime
Index entries for sequences related to palindromes
Index entries for sequences related to compositions

Cf. A002385, A023360, A282584.

nmax = 45; CoefficientList[Series[1/(1 - Sum[Boole[PalindromeQ[k] && PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A002385(k)).

A350850 Members of A350836 that are not in A002385.

Original entry on oeis.org

1, 14, 50, 194, 712, 762, 1100, 1994, 2701, 4959, 58376, 70478, 111538, 116416, 144080, 158736, 712410, 1319216, 1934075, 7709760, 10228166, 11601194, 94663994, 177930006

Offset: 1

J. M. Bergot and Robert Israel, Jan 18 2022

Numbers k that are not palindromic primes, such that the concatenation of the prime factors of k with multiplicity is congruent mod k to the reverse of k.

			a(3) = 50 is a term because A103168(50) = 5 mod 50 = 5 and A340592(50) = 255 mod 50 = 5, but 50 is not a palindromic prime.

Cf. A002385, A103168, A340592, A350836.

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 L, p, i, r;
  L:= sort(map(t -> t[1]$t[2], ifactors(n)[2]));
  r:= L[1];
  for i from 2 to nops(L) do r:= r*10^(1+max(0, ilog10(L[i])))+L[i] od;
  r
end proc:
f(1):= 1:
filter:= proc(n) local r;
r:= revdigs(n);
(f(n) - r) mod n = 0 and (revdigs(n) <> n or not isprime(n))
end proc:
select(filter, [$1..10^6]);

cp's OEIS Frontend

A007500 Primes whose reversal in base 10 is also prime (called "palindromic primes" by David Wells, although that name usually refers to A002385). Also called reversible primes.

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A192139 Powers p^m, m >= 0, of palindromic primes p (A002385).

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A046405 Numbers of the form p*q*r where p,q,r are distinct odd palindromic primes (odd terms from A002385).

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A046485 Sum of first n palindromic primes A002385.

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A046373 Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd palindromic primes (odd terms from A002385).

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Mathematica

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A118064 Decimal expansion of the sum of the reciprocals of the palindromic primes A002385 (Honaker's constant).

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Programs

Mathematica

Formula

Extensions

A287961 Numbers that are the sum of two palindromic primes (A002385).

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Programs

Mathematica

A327428 Record gaps between palindromic primes (lower end): primes A002385(k) where A002385(k+1) - A002385(k) exceeds A002385(j+1) - A002385(j) for all j < k.

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A046405 Numbers of the form pqr where p,q,r are distinct odd palindromic primes (odd terms from A002385).

A046373 Numbers of the form pqr where p,q,r are (not necessarily distinct) odd palindromic primes (odd terms from A002385).