A002385 -id:A002385 - cp's OEIS frontend

A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors.

4, 6, 9, 22, 33, 55, 77, 121, 202, 262, 303, 393, 505, 626, 707, 939, 1111, 1441, 1661, 1991, 3443, 3883, 7997, 10201, 13231, 15251, 18281, 19291, 20602, 22622, 22822, 24842, 26662, 28682, 30903, 31613, 33933, 35653, 37673, 38683, 39993, 60206, 60406, 60806

Offset: 1

Patrick De Geest, Jun 15 1998

Equivalently, semiprime palindromes where both prime factors are palindromes. - Franklin T. Adams-Watters, Apr 11 2011
The sequence "trivially" includes products of palindromic primes p*q where
  a) p = 2 or 3 and q has only digits < 4, as q = 11, 101, 131, 10301, 30103, ...
  b) p <= 11 and q has only digits 0 and 1, as q = 101 and repunit primes A004022
  c) p = 11 and q has only digits spaced out by zeros, as q = 101, 10301, 10501, 10601, 30103, 30203, 30403, 30703, 30803, ... - M. F. Hasler, Jan 04 2022

			The palindrome 35653 is a term since it has 2 factors, 101 and 353, both palindromic.

Lars Blomberg, Table of n, a(n) for n = 1..1000

Cf. A001358 (semiprimes), A002113 (palindromes), A002385 (palindromic primes).
Cf. A046328, A046408.
Subsequence of A188650.

Take[Select[Times@@@Tuples[Select[Prime[Range[5000]],PalindromeQ],2], PalindromeQ]// Union,50] (* Harvey P. Dale, Aug 25 2019 *)

{first(N=50, p=1) = vector(N, i, until( bigomega( p=nxt_A002113(p))==2 && vecmin( apply( is_A002113, factor(p)[,1])),); p)} \\ M. F. Hasler, Jan 04 2022

from sympy import factorint
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal):
    f = factorint(pal)
    return sum(f.values()) == 2 and all(ispal(p) for p in f)
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Aug 14 2022

Intersection of A002113 and A046368; A188649(a(n)) = a(n). - Reinhard Zumkeller, Apr 11 2011

Definition clarified by Franklin T. Adams-Watters, Apr 11 2011

More terms from Lars Blomberg, Nov 06 2015

A070247 Palindromic primes with digit sum 5.

Original entry on oeis.org

5, 131, 10301, 1003001, 100030001, 100111001, 101000010000101, 10000010101000001, 101000000010000000101, 110000000010000000011, 10000000000300000000001, 10000100000100000100001, 100000100000010000001000001, 10000000000000300000000000001, 10000000001000100010000000001

Offset: 1

Amarnath Murthy, May 05 2002

It is conjectured that are just 3 palindromic primes with digit sum 2, namely 2, 11 and 101. If any others exist, they must be of the form 10^(2^k) + 1 with k > 14.
From Jeppe Stig Nielsen, Aug 30 2025: (Start)
It is now known that any additional primes 10^(2^k) + 1 must have k >= 31.
Digit sum 3 yields only one prime, 3, a palindrome in a vacuous way.
Digit sum 4 leads to primes (A062339), but such numbers can never be palindromes. Proof: Let w be any palindrome with digit sum 4. So w = 10^a + 10^b + 10^c + 10^d with a >= b >= c >= d >= 0. But then 10^c + 10^d is a nontrivial divisor of w, showing that w is not prime.
You may have come here searching for the subsequence 5, 131, 10301, 1003001, 100030001, 10000000000300000000001, ... where the largest digit exceeds 1. See A171376 and A100028 for information on them.
(End)

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..386 (all terms below 10^1000; terms n = 1..238 from Chai Wah Wu)
Hans Riesel, Some factors of the numbers Gn = 6^2^n+1 and Hn = 10^2^n+1, Math. Comp. 23 (1969), p. 413-415. With errata reported in Math. Comp. 24 (1970), p. 243.

Cf. A002385, A062341, A070248, A070249.

Do[p = Join[ IntegerDigits[n, 4], Reverse[ Drop[ IntegerDigits[n, 4], -1]]]; q = Plus @@ p; If[q == 5 && PrimeQ[ FromDigits[p]] && q == 5, Print[ FromDigits[p]]], {n, 1, 4 10^8}] (* this coding will not pick up the first entry *)

for(i=0,50,for(j=0,i,p=10^(2*i)+10^(i+j)+10^i+10^(i-j)+1;isprime(p)&&print1(p,", "))) \\ Jeppe Stig Nielsen, Aug 30 2025

Edited by Robert G. Wilson v, May 15 2002

More terms from Chai Wah Wu, Nov 25 2015

A075807 Numbers n such that n-th prime is palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 26, 32, 36, 42, 43, 65, 71, 74, 76, 129, 134, 138, 139, 157, 158, 1263, 1285, 1293, 1367, 1377, 1483, 1519, 1528, 1583, 1635, 1647, 1682, 1726, 1805, 1814, 1867, 1897, 1917, 1928, 2009, 2060, 2083, 2117, 2196, 2250, 2260, 3255, 3267, 3285

Offset: 1

Jani Melik, Oct 13 2002

			26th prime is 101, which is palindromic.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..781 from Vincenzo Librandi, n = 782..5373 from David A. Corneth)

Cf. A000720, A002385, A069217.

[n: n in [1..3500] | Intseq(NthPrime(n), 10) eq Reverse(Intseq(NthPrime(n), 10))]; // Vincenzo Librandi, Mar 24 2019

test := proc(n) local d; d := convert(ithprime(n),base,10); return ListTools[Reverse](d)=d; end; a := []; for n from 1 to 4000 do if test(n) then a := [op(a),n]; end; od; a;

Rest[Flatten[Position[Prime[Range[4000]],?(IntegerDigits[#] == Reverse[ IntegerDigits[#]]&)]]] (* _Harvey P. Dale, Jan 26 2014 *)

isok(n) = my(d=digits(prime(n))); Vecrev(d) == d; \\ Michel Marcus, Mar 24 2019

a(n) = A000720(A002385(n)). - David A. Corneth, Mar 24 2019

Edited by Dean Hickerson, Oct 21 2002

A082806 Palindromes which are prime and the sum of the digits is also prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 131, 151, 191, 313, 353, 373, 757, 797, 919, 10301, 10501, 11311, 12721, 13331, 13931, 14341, 14741, 15551, 16361, 16561, 18181, 19391, 19991, 30103, 30703, 31513, 32323, 33533, 34543, 35153, 35353, 35753, 36563, 38183

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 20 2003

Most of the initial palindromic primes are members.

11 is the only member of even length since the sum of the digits of such palindromes is even and 2 is the only even prime. For the members of odd length the middle digit is odd (except for 2). - Chai Wah Wu, Aug 12 2014

			E.g. 12721 is a palindromic prime and 1+2+7+2+1 = 13 is also prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A002385.

Subsequence of A083393. [Arkadiusz Wesolowski, Sep 14 2011]

N:= 3: # to get all terms of at most 2*N+1 digits
revdigs:= proc(n)
local L,d;
L:= convert(n,base,10);
d:= nops(L);
add(L[i]*10^(d-i),i=1..d);
end proc:
pals:= proc(d)
local x,y;
seq(seq(x*10^(d+1)+y*10^d + revdigs(x),y=0..9),x=10^(d-1)..10^d-1)
end proc;
select(n -> isprime(n) and isprime(convert(convert(n,base,10),`+`)), {2,3,5,7,11,seq(pals(d),d=1..3)}); # Robert Israel, Aug 12 2014

Select[ Range[390000], PrimeQ[ # ] && FromDigits[ Reverse[ IntegerDigits[ # ]]] == # && PrimeQ[ Plus @@ IntegerDigits[ # ]] & ] (* Robert G. Wilson v, Jun 17 2003 *)

from sympy import isprime
A082806 = sorted([n for n in chain(map(lambda x:int(str(x)+str(x)[::-1]),range(1,10**5)),map(lambda x:int(str(x)+str(x)[-2::-1]), range(1,10**5))) if isprime(n) and isprime(sum([int(d) for d in str(n)]))])
# Chai Wah Wu, Aug 12 2014

Corrected and extended by Giovanni Resta, Feb 07 2006

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007

A188649 Least common multiple of reversals of divisors of n in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 84, 31, 574, 255, 488, 71, 162, 91, 20, 84, 22, 32, 168, 260, 62, 72, 1148, 92, 510, 13, 11224, 33, 6106, 1855, 2268, 73, 15106, 93, 40, 14, 6888, 34, 44, 4590, 64, 74, 10248, 658, 260, 1065, 3100, 35, 3240, 55, 149240, 6825, 7820, 95, 7140, 16, 26, 252, 11224, 8680, 66, 76, 12212, 96, 152110, 17, 4536, 37, 6862, 251940, 2024204, 77

Offset: 1

Reinhard Zumkeller, Apr 11 2011

			Divisors(42) = {1,2,3,6,7,14,21,42}, therefore a(42) = lcm(1,2,3,6,7,41,12,24) = 6888.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A003990 (LCM), A027750 (divisors), A004086 (reversal), A188650.

Cf. A000040, A004087, A002385.

a188649 n = foldl lcm 1 $ map a004086 $ filter ((== 0) . mod n) [1..n]

a(n) = lcm(apply(x->fromdigits(Vecrev(digits(x))), divisors(n))); \\ Michel Marcus, Mar 13 2018

from math import lcm
from sympy import divisors
def a(n): return lcm(*(int(str(d)[::-1]) for d in divisors(n)))
print([a(n) for n in range(1, 78)]) # Michael S. Branicky, Aug 14 2022

a(A000040(n)) = A004087(n);

a(A002385(n)) = A002385(n), see A188650 for all fixed points.

A230199 Sum of digits of n-th palindromic prime.

Original entry on oeis.org

2, 3, 5, 7, 2, 2, 5, 7, 10, 11, 7, 11, 13, 14, 16, 19, 22, 23, 19, 20, 5, 7, 8, 7, 8, 10, 13, 14, 11, 16, 17, 13, 17, 16, 17, 14, 17, 19, 20, 20, 25, 19, 22, 23, 28, 29, 7, 8, 10, 13, 14, 8, 13, 13, 14, 17, 19, 22, 16, 17, 19, 23, 20, 23, 22, 25, 22, 23, 29

Offset: 1

Shyam Sunder Gupta, Oct 11 2013

			a(6) =2, since sum of digits of 6th palindromic prime i.e. 101 is 2.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..5953

Cf. A002385, A007953, A007605.

a = {}; Do[z = n*10^(IntegerLength[n] - 1) +
FromDigits@Rest@Reverse@IntegerDigits[n]; If[PrimeQ[z], s = Apply[Plus, IntegerDigits[z]]; AppendTo[a, s]], {n, 1, 10^5}]; Insert[a, 2, 5]
palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Total[ IntegerDigits[ #]]&/@ Select[Prime[Range[5000]],palQ] (* Harvey P. Dale, Oct 10 2014 *)

a(n) = A007953(A002385(n)). - R. J. Mathar, Sep 09 2015

A032595 First of three consecutive palindromes all of which are prime.

Original entry on oeis.org

1878781, 1879781, 1968691, 3589853, 7819187, 108484801, 159181951, 160696061, 175090571, 187090781, 316686613, 319393913, 725585527, 728888827, 731898137, 904080409, 921191129, 930494039, 987191789, 987484789, 10456865401, 10744944701

Offset: 1

Patrick De Geest, May 15 1998

			2,3,5 are palindromes, and consecutive primes, but they are not consecutive palindromes.
2,3,4 are consecutive palindromes, but they are not all primes.
The first time there are three consecutive palindromes which are all primes is 1878781, 1879781, 1880881.  - _N. J. A. Sloane_, Aug 12 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. De Geest, World!Of Palindromic Primes

Cf. A002385, A032596, A032597.

nxt(n)=my(d=digits(n)); i=(#d+1)\2; while(i&&d[i]==9, d[i]=0; d[#d+1-i]=0; i--); if(i, d[i]++; d[#d+1-i]=d[i], d=vector(#d+1); d[1]=d[#d]=1); sum(i=1, #d, 10^(#d-i)*d[i]) \\ From David A. Corneth at A002113
list(lim)=my(v=List(),p=1,q=2,r=3); while(p<=lim\=1, if(isprime(r), if(isprime(q), if(isprime(p), listput(v,p)); p=q; q=r; r=nxt(r), q=nxt(p=r); r=nxt(q)), q=nxt(p=nxt(r)); r=nxt(q))); Vec(v) \\ Charles R Greathouse IV, Aug 11 2021

New name from Charles R Greathouse IV, Aug 11 2021

A046351 Palindromic composite numbers with only palindromic prime factors.

Original entry on oeis.org

4, 6, 8, 9, 22, 33, 44, 55, 66, 77, 88, 99, 121, 202, 242, 252, 262, 303, 343, 363, 393, 404, 484, 505, 525, 606, 616, 626, 686, 707, 808, 909, 939, 1111, 1331, 1441, 1661, 1991, 2112, 2222, 2662, 2772, 2882, 3333, 3443, 3773, 3883, 3993, 4224, 4444, 5445

Offset: 1

Patrick De Geest, Jun 15 1998

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A002385, A032350, A033620, A046349, A046350.

palQ[n_]:=Reverse[x=IntegerDigits[n]]==x; Select[Range[4,5500],!PrimeQ[#]&&And@@palQ/@Join[{#},First/@FactorInteger[#]]&](* Jayanta Basu, Jun 05 2013 *)

from itertools import product
from sympy import isprime, primefactors as pf
def pal(n): s = str(n); return s == s[::-1]
def palsthru(maxdigits):
  midrange = [[""], [str(i) for i in range(10)]]
  for digits in range(1, maxdigits+1):
    for p in product("0123456789", repeat=digits//2):
      left = "".join(p)
      if len(left) and left[0] == '0': continue
      for middle in midrange[digits%2]: yield int(left+middle+left[::-1])
def okpal(p): return p > 3 and not isprime(p) and all(pal(f) for f in pf(p))
print(list(filter(okpal, palsthru(4)))) # Michael S. Branicky, Apr 06 2021

(A032350 INTERSECT A033620) MINUS {1}. - R. J. Mathar, Sep 09 2015

A054217 Primes p with property that p concatenated with its emirp p' (prime reversal) forms a palindromic prime of the form 'primemirp' (rightmost digit of p and leftmost digit of p' are blended together - p and p' palindromic allowed).

Original entry on oeis.org

2, 3, 5, 7, 13, 31, 37, 79, 113, 179, 181, 199, 353, 727, 787, 907, 937, 967, 983, 1153, 1193, 1201, 1409, 1583, 1597, 1657, 1831, 1879, 3083, 3089, 3319, 3343, 3391, 3541, 3643, 3853, 7057, 7177, 7507, 7681, 7867, 7949, 9103, 9127, 9173, 9209, 9439, 9547, 9601

Offset: 1

Patrick De Geest, Feb 15 2000

Original idea from G. L. Honaker, Jr..

			E.g., prime 113 has emirp 311 and 11311 is a palindromic prime, so 113 is a term.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A054218, A000040, A006567, A048054, A002385.

empQ[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];And@@PrimeQ[ {FromDigits[ rev],FromDigits[Join[Most[idn],rev]]}]]; Select[Prime[ Range[ 1200]],empQ] (* Harvey P. Dale, Mar 26 2013 *)

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n); srev = s[::-1]
    return isprime(int(srev)) and isprime(int(s[:-1] + srev))
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Nov 17 2023

Corrected (a(30)=3089 inserted) by Harvey P. Dale, Mar 26 2013

A070248 Palindromic primes with digit sum 7.

Original entry on oeis.org

7, 151, 313, 10501, 11311, 30103, 1201021, 3001003, 100050001, 100131001, 101030101, 110111011, 111010111, 10000500001, 1100011100011, 1100101010011, 100020010020001

Offset: 1

Amarnath Murthy, May 05 2002

Chai Wah Wu, Table of n, a(n) for n = 1..1111

Cf. A002385, A070247 & A070249.

Do[p = Join[ IntegerDigits[n], Reverse[ Drop[ IntegerDigits[n], -1]]]; q = Plus @@ p; If[ PrimeQ[ FromDigits[p]] && q == 7, Print[ FromDigits[p]]], {n, 1, 10^7}]

Edited by Robert G. Wilson v, May 15 2002

cp's OEIS Frontend

A046376 Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors.

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A070247 Palindromic primes with digit sum 5.

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A075807 Numbers n such that n-th prime is palindromic.

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A082806 Palindromes which are prime and the sum of the digits is also prime.

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Maple

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A188649 Least common multiple of reversals of divisors of n in decimal representation.

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Haskell

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A230199 Sum of digits of n-th palindromic prime.

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A032595 First of three consecutive palindromes all of which are prime.

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