A002385 -id:A002385 - cp's OEIS frontend

A070249 Palindromic primes with digit sum 8.

10601, 11411, 30203, 31013, 1022201, 1120211, 1300031, 3002003, 100060001, 103000301, 111020111, 300020003, 300101003, 10002220001, 10200200201, 10210001201, 1000030300001, 1021000001201, 1030000000301, 1101010101011

Offset: 1

Amarnath Murthy, May 05 2002

Chai Wah Wu, Table of n, a(n) for n = 1..1757

Cf. A002385, A070247 and A070248.

Do[p = Join[ IntegerDigits[n], Reverse[ Drop[ IntegerDigits[n], -1]]]; q = Plus @@ p; If[ PrimeQ[ FromDigits[p]] && q == 8, Print[ FromDigits[p]]], {n, 1, 10^7}]
Select[Prime[Range[1626*10^4]],Total[IntegerDigits[#]]==8&&PalindromeQ[#]&] (* The program generates the first 13 terms of the sequence. *) (* Harvey P. Dale, Jul 18 2022 *)

Edited by Robert G. Wilson v, May 15 2002

A082584 Fractal palindromic primes of first order.

Original entry on oeis.org

313, 353, 373, 383, 727, 757, 787, 797, 11311, 11411, 1311131, 1317131, 1513151, 1917191, 9196919, 9199919, 10301110301, 10301910301, 10501210501, 10501910501, 10601110601, 12421212421, 12421812421, 12721612721, 13331113331, 13331713331, 13831913831, 13931413931

Offset: 1

Lekraj Beedassy, May 07 2003

A first order fractal palindromic prime is one of the form WmW, where either wing W about a central digit m, is itself a palindromic prime, which, however, may not be further split in this manner to maintain the property.

[Log_10(a(n))] must be of the form k2. - Robert G. Wilson v, Jul 22 2005

			12421812421 is in the sequence because it is a concatenation of the palindromic prime part 12421 with itself, hinging over the central 8;12421 cannot however be split into simpler palindromic primes in this way.

Michael S. Branicky, Table of n, a(n) for n = 1..21046 (all terms < 10^26, corresponding to b-file of A082598)

Cf. A002385.

f[n_] := Block[{m = n}, pd = IntegerDigits[m]; pd = Take[pd, Floor[Length[pd]/2]]; If[PrimeQ[m] && PrimeQ[FromDigits[pd]] && pd == Reverse[pd] && m == FromDigits[Reverse[IntegerDigits[m]]] && ! f[FromDigits[pd]], True, False]]; Do[ If[ f[n], Print[n]], {n, 10^10}] (* Robert G. Wilson v, Jul 22 2005 *)

a(25) and beyond from Michael S. Branicky, May 19 2024

A082620 a(1) = 1, then the smallest palindromic prime obtained by inserting digits anywhere in a(n-1).

Original entry on oeis.org

1, 11, 101, 10301, 1003001, 100030001, 10003630001, 1000136310001, 100010363010001, 10001036363010001, 1000103639363010001, 100010356393653010001, 10001033563936533010001, 1000103305639365033010001, 100010313056393650313010001, 10001031305636963650313010001

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 29 2003

Giovanni Resta, Table of n, a(n) for n = 1..92

Cf. A002385, A082621, A082622, A082623, A082624.

Corrected by R. J. Mathar, Oct 01 2006
a(7)-a(10) from Felix Fröhlich, Oct 16 2014
a(11)-a(12) from Felix Fröhlich, Nov 26 2014
a(13)-a(16) from Felix Fröhlich, Apr 02 2015
Terms a(8)-a(16) corrected by Giovanni Resta, Sep 20 2019

A082621 a(1) = 2, then the smallest palindromic prime obtained by inserting digits anywhere in a(n-1) (including at the ends).

Original entry on oeis.org

2, 727, 37273, 3072703, 307323703, 30073237003, 3006732376003, 300067323760003, 30000673237600003, 3000067382837600003, 300006738242837600003, 30000673820402837600003, 3000063738204028373600003, 300006373821040128373600003, 30000635738210401283753600003

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 29 2003

Giovanni Resta, Table of n, a(n) for n = 1..51

Cf. A002385, A082620, A082622, A082623, A082624.

Corrected by R. J. Mathar, Oct 01 2006

More terms from Giovanni Resta, Sep 20 2019

A083185 Palindromic primes using only nonprime digits (0,1,4,6,8,9).

Original entry on oeis.org

11, 101, 181, 191, 919, 10601, 11411, 16061, 16661, 18181, 18481, 19891, 19991, 91019, 94049, 94649, 94849, 94949, 96469, 98689, 1008001, 1114111, 1160611, 1180811, 1186811, 1190911, 1196911, 1409041, 1411141, 1444441, 1461641

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 26 2003

Harvey P. Dale, Table of n, a(n) for n = 1..100
P. De Geest, World!Of Palindromic Primes

Cf. A045336, A019546, A002385.

Palindromes in A034844.

Select[ Prime[ Range[111500]], IntegerDigits[ # ] == Reverse[ IntegerDigits[ # ]] && Union[ Join[ IntegerDigits[ # ], {0, 1, 4, 6, 8, 9}]] == {0, 1, 4, 6, 8, 9} & ]
Select[Prime[Range[120000]],PalindromeQ[#]&&NoneTrue[IntegerDigits[#], PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 08 2018 *)

Edited and extended by Patrick De Geest, Jun 11 2003

A088269 Palindromic primes that yield a prime when sandwiched between two 1's. (Prefixing and suffixing a 1 on both sides yields another palindromic prime.)

Original entry on oeis.org

3, 5, 131, 383, 797, 11411, 16061, 16361, 19391, 33533, 36263, 73037, 75557, 79397, 1074701, 1126211, 1145411, 1175711, 1221221, 1243421, 1287821, 1303031, 1311131, 1328231, 1363631, 1489841, 1579751, 1600061, 1707071, 1748471

Offset: 1

Amarnath Murthy, Sep 28 2003

There are two 1-digit terms, three 3-digit terms, nine 5-digit terms, 93 7-digit terms, 241 9-digit terms and no terms with an even number of digits. - Zak Seidov, Feb 23 2005

			Take palindromic primes (A002385) and see whether inserting them between two digits '1' again yields a prime:
Insert a(1) = 3 between the digits of 11 to get 131, a prime.
Insert a(2) = 5 between the digits of 11 to get 151, a prime.
Inserting 11 between two '1's yields 1111 = 11 * 101, not a prime.
Insert a(3) = 131 between the digits of 11 to get 11311, a prime.
Insert a(10) = 33533 between the digits of 11 to get 1335331, a prime, etc.
797 is a term as 17971 is also a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..100

Cf. A088270, A088271, A088272.

Cf. A002385.

Do[If[PrimeQ[n] && Reverse[IntegerDigits[n]] == IntegerDigits[n] && PrimeQ[ToExpression["1" <> ToString[n*10+1]]], Print[n]], {n, 1, 2*10^6}] (* Ryan Propper, Jul 09 2005 *)
palsQ[n_]:=Module[{idn=IntegerDigits[n],idn1},idn1=Join[{1},idn,{1}]; idn==Reverse[idn]&&idn1==Reverse[idn1]&&PrimeQ[FromDigits[idn1]]]; Select[Prime[Range[150000]],palsQ] (* Harvey P. Dale, Jan 04 2012 *)

is_A088269(n)={isprime(n)&&(n=digits(n))==Vecrev(n)&&isprime(fromdigits(concat([1,n,1])))} \\ M. F. Hasler, Nov 19 2018

a(6)-a(30) from Ryan Propper, Jul 09 2005
Entry revised by N. J. A. Sloane, Apr 29 2007
Edited by M. F. Hasler, Nov 19 2018

A128375 Palindromic primes that start and end with 9.

Original entry on oeis.org

919, 929, 90709, 91019, 93139, 93239, 93739, 94049, 94349, 94649, 94849, 94949, 95959, 96269, 96469, 96769, 97379, 97579, 97879, 98389, 98689, 9002009, 9015109, 9024209, 9037309, 9042409, 9043409, 9045409, 9046409, 9049409, 9067609, 9073709, 9076709, 9078709

Offset: 1

Lekraj Beedassy, Feb 27 2007

Subsequence of A002385 (palindromic primes) and A002113 (palindromic numbers). Complement of A222729 (palindromic composite numbers starting with a digit 9) with respect to A043044 (palindromic numbers starting with a digit 9). - Jaroslav Krizek, Mar 09 2013

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A002385, A128374, A128376.

Select[Prime[Range[607000]], Reverse[x = IntegerDigits[#]] == x && First[x] == 9 &] (* Jayanta Basu, Jun 24 2013 *)

Extended by Ray Chandler, Feb 28 2007

A188650 Fixed points of A188649: numbers divisible by the reverse of all their divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 33, 40, 44, 55, 66, 77, 88, 99, 101, 110, 121, 131, 151, 181, 191, 202, 220, 242, 262, 303, 313, 353, 363, 373, 383, 393, 404, 440, 484, 505, 606, 626, 707, 727, 757, 787, 797, 808, 909, 919, 929, 939, 1010, 1111, 1331, 1441, 1661, 1991, 2020, 2222, 2662, 2882, 3333, 3443, 3883, 3993, 4040, 4444, 5555, 6666, 6886, 7777, 7997, 8888

Offset: 1

Reinhard Zumkeller, Apr 11 2011

A188649(a(n)) = a(n);

A002385 and A046376 are subsequences; subsequence of A002113.

Michael S. Branicky, Table of n, a(n) for n = 1..10700

import Data.List (elemIndices)
a188650 n = a188650_list !! (n-1)
a188650_list =
   map succ $ elemIndices 0 $ zipWith (-) [1..] $ map a188649 [1..]

rev(n:int,B=10)=my(m=n%B);n\=B;while(n>0,m=m*B+n%B;n\=B);m
is(n)=fordiv(n,d,if(n%rev(d),return(0)));1 \\ Charles R Greathouse IV, Jul 14 2011

from math import lcm
from sympy import divisors
def ok(n): return n == lcm(*(int(str(d)[::-1]) for d in divisors(n)))
print([k for k in range(1, 10000) if ok(k)]) # Michael S. Branicky, Sep 30 2022

A192137 Numbers m such that their concatenation of prime divisors are palindromic numbers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 16, 25, 27, 32, 39, 49, 64, 69, 81, 101, 117, 119, 121, 125, 128, 129, 131, 151, 159, 181, 191, 207, 219, 243, 249, 256, 259, 313, 329, 339, 343, 351, 353

Offset: 1

Jaroslav Krizek, Jun 24 2011

The corresponding values of palindromic concatenation in A192138. Superset of A002385 (palindromic primes), A192139 and A192140.

			Concatenation of prime divisors of 39 = 3 * 13 is 313 (palindromic number).

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

Cf. A002113, A002385, A084317, A084092, A192138, A192139, A192140, A192141.

Select[Range[2,500],PalindromeQ[FromDigits[Flatten[IntegerDigits/@ FactorInteger[ #][[All,1]]]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 02 2017 *)

A383815 Palindromic primes in A380943.

Original entry on oeis.org

313, 373, 797, 11311, 13331, 13931, 17971, 19991, 31013, 35353, 36263, 36563, 38783, 71317, 79397, 97379, 98389, 1129211, 1196911, 1611161, 1793971, 1982891, 3106013, 3166613, 3193913, 3236323, 3288823, 3304033, 3319133, 3329233, 3365633, 3417143, 3447443, 3449443, 3515153, 3670763

Offset: 1

James C. McMahon and Robert G. Wilson v, Jun 06 2025

A380943 requires that primes, p_n, written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

Intersection of A002385 and A380943.

			The palindromic prime 313 is formed by the concatenation of the primes 31 and 3, which reversed, also form the prime 331. The palindromic prime 13931 is formed by the concatenation of 139 and 31; 31139 is also prime.

Cf. A000040, A002385, A105184, A380943, A383810, A383811, A383812, A383813.

rev:= proc(n) local L,i;
   L:= convert(n,base,10);
   add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(x,y) y + 10^(ilog10(y)+1)*x end proc:
filter:= proc(z) local i,x,y;
  if not isprime(z) then return false fi;
  for i from 1 to ilog10(z) do
    x:= z mod 10^i;
    if x < 10^(i-1) then next fi;
    y:= (z-x)/10^i;
    if isprime(x) and isprime(y) and isprime(tcat(x,y)) then return true fi;
  od;
  false
end proc:
N:= 7: # for terms of up to 7 digits
R:= NULL:
for d from 1 to (N-1)/2 do
  for x from 10^(d-1) to 10^d-1 do
    for y from 0 to 9 do
      z:= rev(x) + 10^d * y + 10^(d+1)*x;
      if filter(z) then R:= R,z fi
od od od:
R;  # Robert Israel, Jun 08 2025

f[n_] := Block[{cnt = 0, id = IntegerDigits@ n, k = 1, len, p, q, qp}, len = Length@ id; While[k < len, p = Take[id, k]; q = Take[id, -len + k]; qp = FromDigits[Join[q, p]]; If[ PrimeQ@ FromDigits@ p && PrimeQ@ FromDigits@ q && PrimeQ@ qp && IntegerLength@ qp == len, cnt++]; k++]; cnt]; fQ[n_] := Reverse[idn = IntegerDigits@ n] == idn && f@ n > 0; Select[ Prime@ Range@ 264000, fQ]

cp's OEIS Frontend

A070249 Palindromic primes with digit sum 8.

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A082584 Fractal palindromic primes of first order.

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A082620 a(1) = 1, then the smallest palindromic prime obtained by inserting digits anywhere in a(n-1).

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A082621 a(1) = 2, then the smallest palindromic prime obtained by inserting digits anywhere in a(n-1) (including at the ends).

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A083185 Palindromic primes using only nonprime digits (0,1,4,6,8,9).

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A088269 Palindromic primes that yield a prime when sandwiched between two 1's. (Prefixing and suffixing a 1 on both sides yields another palindromic prime.)

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Mathematica

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A128375 Palindromic primes that start and end with 9.

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A188650 Fixed points of A188649: numbers divisible by the reverse of all their divisors.

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A192137 Numbers m such that their concatenation of prime divisors are palindromic numbers.

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