A033620 -id:A033620 - cp's OEIS frontend

A002385 Palindromic primes: prime numbers whose decimal expansion is a palindrome.

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741, 15451, 15551, 16061, 16361, 16561, 16661, 17471, 17971, 18181, 18481, 19391, 19891, 19991

Offset: 1

N. J. A. Sloane, Simon Plouffe

Every palindrome with an even number of digits is divisible by 11, so 11 is the only member of the sequence with an even number of digits. - David Wasserman, Sep 09 2004
This holds in any number base A006093(n), n>1. - Lekraj Beedassy, Mar 07 2005 and Dec 06 2009
The log-log plot shows the fairly regular structure of these numbers. - T. D. Noe, Jul 09 2013
Conjecture: The only primes with palindromic prime indices that are palindromic primes themselves are 3, 5 and 11. Tested for the primes with the first 8000000 palindromic prime indices. - Ivan N. Ianakiev, Oct 10 2014
It follows from the above conjecture that 2 is the only k such that k, prime(k), prime(m) = k + prime(k) and m are all palindromic primes. - Ivan N. Ianakiev, Mar 17 2025
Banks, Hart, and Sakata derive a nontrivial upper bound for the number of prime palindromes n <= x as x -> oo. It follows that almost all palindromes are composite. The results hold in any base. The authors use Weil's bound for Kloosterman sums. - Jonathan Sondow, Jan 02 2018
Number of terms < 100^k, k >= 1: 5, 20, 113, 781, 5953, 47995, 401698, .... - Robert G. Wilson v, Jan 03 2018, corrected by M. F. Hasler, Dec 19 2024
Initially the above comment listed 4, 20, 113, ... which is the number of terms less than 10, 1000, 10^5, ..., i.e., up to 10^(2k-1), k >= 1. The number of terms < 10^k are the cumulative sums of A016115(n) (number of prime palindromes with n digits) up to n = k. - M. F. Hasler, Dec 19 2024

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 228.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 120-121.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..47995 (all palindromic primes with fewer than 12 digits), Oct 14 2015, extending earlier b-files from T. D. Noe and A. Olah.
W. D. Banks, D. N. Hart, and M. Sakata, Almost all palindromes are composite, Math. Res. Lett., 11 No. 5-6 (2004), 853-868.
K. S. Brown, On General Palindromic Numbers
C. K. Caldwell, "Top Twenty" page, Palindrome
Patrick De Geest, World!Of Palindromic Primes
Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020. Mentions this sequence.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 10, 18.
T. D. Noe, Log-log plot of the first 401696 terms
I. Peterson, Math Trek, Palindromic Primes
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Reciprocal sum of palindromes, arXiv:1803.00161 [math.CA], 2018.
M. Shafer, First 401066 Palprimes
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Palindromic Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Wikipedia, Palindromic prime
Index entries for sequences related to palindromes

A007500 = this sequence union A006567.
Subsequence of A188650; A188649(a(n)) = a(n); see A033620 for multiplicative closure. [Reinhard Zumkeller, Apr 11 2011]
Cf. A016041, A029732, A069469, A117697, A046942, A032350 (Palindromic nonprime numbers).
Cf. A016115 (number of prime palindromes with n digits).

Filtered([1..20000],n->IsPrime(n) and ListOfDigits(n)=Reversed(ListOfDigits(n))); # Muniru A Asiru, Mar 08 2019

a002385 n = a002385_list !! (n-1)
a002385_list = filter ((== 1) . a136522) a000040_list
-- Reinhard Zumkeller, Apr 11 2011

ff := proc(n) local i,j,k,s,aa,nn,bb,flag; s := n; aa := convert(s,string); nn := length(aa); bb := ``; for i from nn by -1 to 1 do bb := cat(bb,substring(aa,i..i)); od; flag := 0; for j from 1 to nn do if substring(aa,j..j)<>substring(bb,j..j) then flag := 1 fi; od; RETURN(flag); end; gg := proc(i) if ff(ithprime(i)) = 0 then RETURN(ithprime(i)) fi end;
rev:=proc(n) local nn, nnn: nn:=convert(n,base,10): add(nn[nops(nn)+1-j]*10^(j-1),j=1..nops(nn)) end: a:=proc(n) if n=rev(n) and isprime(n)=true then n else fi end: seq(a(n),n=1..20000); # rev is a Maple program to revert a number - Emeric Deutsch, Mar 25 2007
# A002385 Gets all base-10 palindromic primes with exactly d digits, in the list "Res"
d:=7; # (say)
if d=1 then Res:= [2,3,5,7]:
elif d=2 then Res:= [11]:
elif d::even then
    Res:=[]:
else
    m:= (d-1)/2:
    Res2 := [seq(seq(n*10^(m+1)+y*10^m+digrev(n), y=0..9), n=10^(m-1)..10^m-1)]:
    Res:=[]: for x in Res2 do if isprime(x) then Res:=[op(Res),x]; fi: od:
fi:
Res; # N. J. A. Sloane, Oct 18 2015

Select[ Prime[ Range[2100] ], IntegerDigits[#] == Reverse[ IntegerDigits[#] ] & ]
lst = {}; e = 3; Do[p = n*10^(IntegerLength[n] - 1) + FromDigits@Rest@Reverse@IntegerDigits[n]; If[PrimeQ[p], AppendTo[lst, p]], {n, 10^e - 1}]; Insert[lst, 11, 5] (* Arkadiusz Wesolowski, May 04 2012 *)
Join[{2,3,5,7,11},Flatten[Table[Select[Prime[Range[PrimePi[ 10^(2n)]+1, PrimePi[ 10^(2n+1)]]],# == IntegerReverse[#]&],{n,3}]]] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Apr 22 2016 *)
genPal[n_Integer, base_Integer: 10] := Block[{id = IntegerDigits[n, base], insert = Join[{{}}, {# - 1} & /@ Range[base]]}, FromDigits[#, base] & /@ (Join[id, #, Reverse@id] & /@ insert)]; k = 1; lst = {2, 3, 5, 7}; While[k < 19, p = Select[genPal[k], PrimeQ];
If[p != {}, AppendTo[lst, p]]; k++]; Flatten@ lst (* RGWv *)
Select[ Prime[ Range[2100]], PalindromeQ] (* Jean-François Alcover, Feb 17 2018 *)
NestList[NestWhile[NextPrime, #, ! PalindromeQ[#2] &, 2] &, 2, 41] (* Jan Mangaldan, Jul 01 2020 *)

is(n)=n==eval(concat(Vecrev(Str(n))))&&isprime(n) \\ Charles R Greathouse IV, Nov 20 2012

forprime(p=2,10^5, my(d=digits(p,10)); if(d==Vecrev(d),print1(p,", "))); \\ Joerg Arndt, Aug 17 2014

A002385_row(n)=select(is_A002113, primes([10^(n-1),10^n])) \\ Terms with n digits. For larger n, better filter primes in palindromes. - M. F. Hasler, Dec 19 2024

from itertools import chain
from sympy import isprime
A002385 = sorted((n for n in chain((int(str(x)+str(x)[::-1]) for x in range(1,10**5)),(int(str(x)+str(x)[-2::-1]) for x in range(1,10**5))) if isprime(n))) # Chai Wah Wu, Aug 16 2014

from sympy import isprime
A002385 = [*filter(isprime, (int(str(x) + str(x)[-2::-1]) for x in range(10**5)))]
A002385.insert(4, 11)  # Yunhan Shi, Mar 03 2023

from sympy import isprime
from itertools import count, islice, product
def A002385gen(): # generator of palprimes
    yield from [2, 3, 5, 7, 11]
    for d in count(3, 2):
        for last in "1379":
            for p in product("0123456789", repeat=d//2-1):
                left = "".join(p)
                for mid in [[""], "0123456789"][d&1]:
                    t = int(last + left + mid + left[::-1] + last)
                    if isprime(t):
                        yield t
print(list(islice(A002385gen(), 46))) # Michael S. Branicky, Apr 13 2025

[n for n in (2..18181) if is_prime(n) and Word(n.digits()).is_palindrome()] # Peter Luschny, Sep 13 2018

Intersection of A000040 (primes) and A002113 (palindromes).
A010051(a(n)) * A136522(a(n)) = 1. [Reinhard Zumkeller, Apr 11 2011]
Complement of A032350 in A002113. - Jonathan Sondow, Jan 02 2018

More terms from Larry Reeves (larryr(AT)acm.org), Oct 25 2000

Comment from A006093 moved here by Franklin T. Adams-Watters, Dec 03 2009

A051038 11-smooth numbers: numbers whose prime divisors are all <= 11.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140

Offset: 1

Eric W. Weisstein

A155182 is a finite subsequence. - Reinhard Zumkeller, Jan 21 2009
From Federico Provvedi, Jul 09 2022: (Start)
In general, if p=A000040(k) is the k-th prime, with k>1, p-smooth numbers are also those positive integers m such that A000010(A002110(k))*m == A000010(A002110(k)*m).
With k=5, p = A000040(5) = 11, the primorial p# = A002110(5) = 2310, and its Euler totient is A000010(2310) = 480, so the 11-smooth numbers are also those positive integers m such that 480*m == A000010(2310*m). (End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (First 5000 terms from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Smooth Number.

Subsequence of A033620.
For p-smooth numbers with other values of p, see A003586, A051037, A002473, A080197, A080681, A080682, A080683.
A000010, A002110.

[n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(11)]; // Bruno Berselli, Sep 24 2012

mx = 150; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}] (* Robert G. Wilson v, Aug 17 2012 *)
aQ[n_]:=Max[First/@FactorInteger[n]]<=11; Select[Range[140],aQ[#]&] (* Jayanta Basu, Jun 05 2013 *)
Block[{k=5,primorial:=Times@@Prime@Range@#&},Select[Range@200,#*EulerPhi@primorial@k==EulerPhi[#*primorial@k]&]] (* Federico Provvedi, Jul 09 2022 *)

test(n)=m=n; forprime(p=2,11, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

list(lim,p=11)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=11): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 67))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A051038(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,11)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 11} p/(p-1) = (2*3*5*7*11)/(1*2*4*6*10) = 77/16. - Amiram Eldar, Sep 22 2020

A046368 Products of two palindromic primes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121, 202, 262, 302, 303, 362, 382, 393, 453, 505, 543, 573, 626, 655, 706, 707, 746, 755, 766, 905, 917, 939, 955, 1057, 1059, 1111, 1119, 1149, 1267, 1337, 1441, 1454, 1514, 1565, 1574, 1594, 1661, 1765, 1838, 1858

Offset: 1

Patrick De Geest, Jun 15 1998

Intersection of A001358 and A033620; A046368 is a subsequence. - Reinhard Zumkeller, Apr 10 2011
Equivalently, semiprimes where both prime factors are palindromes. - Franklin T. Adams-Watters, Apr 11 2011
See A046376 for the subsequence of palindromic terms. - M. F. Hasler, Jan 04 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A001358 (semiprimes), A002113 (palindromes), A002385 (palindromic primes), A046376 (subsequence of palindromes), A046400.

palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Select[Range[1800], PrimeOmega[#] == 2 && And @@ palQ /@ First /@ FactorInteger[#] &] (* Jayanta Basu, Jun 23 2013 *)

select( {is_A046368(n)=bigomega(n)==2 && vecmin( apply( is_A002113, factor(n)[, 1]))}, [1..9999]) \\ M. F. Hasler, Jan 04 2022

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

Definition simplified by M. F. Hasler, Jan 04 2022

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

A046349 Composite numbers with only palindromic prime factors.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 49, 50, 54, 55, 56, 60, 63, 64, 66, 70, 72, 75, 77, 80, 81, 84, 88, 90, 96, 98, 99, 100, 105, 108, 110, 112, 120, 121, 125, 126, 128, 132, 135, 140, 144, 147, 150

Offset: 1

Patrick De Geest, Jun 15 1998

Robert Israel, Table of n, a(n) for n = 1..3020

Cf. A002385, A033620, A046350, A046351.

isA046349 := proc(n)
    simplify(isA033620(n) and not isprime(n)) ;
end proc:
for n from 2 to 300 do
    if isA046349(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 09 2015

palQ[n_]:=Reverse[x=IntegerDigits[n]]==x; Select[Range[4,150],!PrimeQ[#]&&And@@palQ/@First/@FactorInteger[#]&] (* Jayanta Basu, Jun 05 2013 *)
Select[Range[200],CompositeQ[#]&&AllTrue[FactorInteger[#][[All,1]],PalindromeQ]&] (* Harvey P. Dale, May 15 2022 *)

from sympy import isprime, primefactors
def pal(n): s = str(n); return s == s[::-1]
def ok(n): return not isprime(n) and all(pal(f) for f in primefactors(n))
print(list(filter(ok, range(4, 151)))) # Michael S. Branicky, Apr 06 2021

A033620 INTERSECT A002808. - R. J. Mathar, Sep 09 2015

A046402 Numbers with exactly 4 distinct palindromic prime factors.

Original entry on oeis.org

210, 330, 462, 770, 1155, 3030, 3930, 4242, 4530, 5430, 5502, 5730, 6342, 6666, 7070, 7602, 8022, 8646, 9170, 9390, 9966, 10570, 10590, 10605, 11110, 11190, 11490, 11946, 12606, 12670, 13146, 13370, 13755, 14410, 14826, 15554, 15666, 15855, 16086, 16610, 16665

Offset: 1

Patrick De Geest, Jun 15 1998

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

Intersection of A033620 and A046386.

Cf. A046370.

Take[Times@@@Module[{nn=300,pp},pp=Select[Prime[Range[nn]],PalindromeQ];Subsets[pp,{4}]]//Union,40] (* Harvey P. Dale, Aug 13 2024 *)

Offset changed and a(37) onwards from Andrew Howroyd, Aug 14 2024

A046404 Odd numbers with exactly 2 distinct palindromic prime factors.

Original entry on oeis.org

15, 21, 33, 35, 55, 77, 303, 393, 453, 505, 543, 573, 655, 707, 755, 905, 917, 939, 955, 1057, 1059, 1111, 1119, 1149, 1267, 1337, 1441, 1565, 1661, 1765, 1865, 1915, 1991, 2101, 2181, 2191, 2271, 2361, 2391, 2471, 2611, 2681, 2757, 2787, 3443, 3635, 3785

Offset: 1

Patrick De Geest, Jun 15 1998

Harvey P. Dale, Table of n, a(n) for n = 1..501

Intersection of A033620 and A046388.

Cf. A046372.

dpf2Q[n_]:=Module[{pfs=Transpose[FactorInteger[n]][[1]],a,b},a=First[pfs];b=Last[pfs];PrimeNu[n]==PrimeOmega[n]==2 && a==FromDigits[Reverse[ IntegerDigits[ a]]] && b==FromDigits[Reverse[ IntegerDigits[b]]]]; Select[ Range[1,3801,2],dpf2Q] (* Harvey P. Dale, Jul 14 2013 *)

Offset changed by Andrew Howroyd, Aug 14 2024

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

A046350 Odd composite numbers with only palindromic prime factors.

Original entry on oeis.org

9, 15, 21, 25, 27, 33, 35, 45, 49, 55, 63, 75, 77, 81, 99, 105, 121, 125, 135, 147, 165, 175, 189, 225, 231, 243, 245, 275, 297, 303, 315, 343, 363, 375, 385, 393, 405, 441, 453, 495, 505, 525, 539, 543, 567, 573, 605, 625, 655, 675, 693, 707, 729, 735, 755

Offset: 1

Patrick De Geest, Jun 15 1998

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

Cf. A002385, A033620, A046349, A046351.

palQ[n_]:=Reverse[x=IntegerDigits[n]]==x; Select[Range[9,755,2],!PrimeQ[#]&&And@@palQ/@First/@FactorInteger[#]&] (* Jayanta Basu, Jun 05 2013 *)
Select[Range[9,800,2],CompositeQ[#]&&AllTrue[FactorInteger[#][[All,1]], PalindromeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 08 2018 *)

from sympy import isprime, primefactors
def pal(n): s = str(n); return s == s[::-1]
def ok(n): return not isprime(n) and all(pal(f) for f in primefactors(n))
print(list(filter(ok, range(9, 756, 2)))) # Michael S. Branicky, Apr 06 2021

A046406 Odd numbers with exactly 4 distinct palindromic prime factors.

Original entry on oeis.org

1155, 10605, 13755, 15855, 16665, 19005, 20055, 21615, 23331, 24915, 29865, 30261, 31515, 32865, 34881, 37065, 38885, 39165, 40215, 41811, 44121, 50435, 51645, 58135, 58245, 61545, 63195, 69685, 72303, 73535, 76335, 79485, 81543

Offset: 1

Patrick De Geest, Jun 15 1998

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

Intersection of A033620 and A046390.

Cf. A046374.

Offset changed by Andrew Howroyd, Aug 14 2024

cp's OEIS Frontend

A002385 Palindromic primes: prime numbers whose decimal expansion is a palindrome.

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A051038 11-smooth numbers: numbers whose prime divisors are all <= 11.

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A046368 Products of two palindromic primes.

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A046351 Palindromic composite numbers with only palindromic prime factors.

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A046349 Composite numbers with only palindromic prime factors.

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A046402 Numbers with exactly 4 distinct palindromic prime factors.

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A046404 Odd numbers with exactly 2 distinct palindromic prime factors.

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