A188649 -id:A188649 - cp's OEIS frontend

A004086 Read n backwards (referred to as R(n) in many sequences).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 6, 16, 26, 36, 46, 56, 66, 76, 86, 96, 7, 17, 27, 37, 47

Offset: 0

N. J. A. Sloane

Also called digit reversal of n.

Leading zeros (after the reversal has taken place) are omitted. - N. J. A. Sloane, Jan 23 2017

Indranil Ghosh, Table of n, a(n) for n = 0..50000 (first 1001 terms from Franklin T. Adams-Watters)
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], Dec 08 2020.
Michael Penn, A digit moving number puzzle., YouTube video, 2022.

Cf. A004185, A004186, A009996, A073138, A188649, A221714.

Cf. A030101, A030102, A030103, A030104, A030105, A030107, A030108.

a004086 = read . reverse . show  -- Reinhard Zumkeller, Apr 11 2011

|.&.": i.@- 1e5 NB. Stephen Makdisi, May 14 2018

read transforms; A004086 := digrev; #cf "Transforms" link at bottom of page
A004086:=proc(n) local s,t; if n<10 then n else s:=irem(n,10,'t'); while t>9 do s:=s*10+irem(t,10,'t') od: s*10+t fi end; # M. F. Hasler, Jan 29 2012

Table[FromDigits[Reverse[IntegerDigits[n]]], {n, 0, 75}]
IntegerReverse[Range[0,80]](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2018 *)

dig(n) = {local(m=n,r=[]); while(m>0,r=concat(m%10,r);m=floor(m/10));r}
A004086(n) = {local(b,m,r);r=0;b=1;m=dig(n);for(i=1,matsize(m)[2],r=r+b*m[i];b=b*10);r} \\ Michael B. Porter, Oct 16 2009

A004086(n)=fromdigits(Vecrev(digits(n))) \\ M. F. Hasler, Nov 11 2010, updated May 11 2015, Sep 13 2019

def A004086(n):
    return int(str(n)[::-1]) # Chai Wah Wu, Aug 30 2014

For n > 0, a(a(n)) = n iff n mod 10 != 0. - Reinhard Zumkeller, Mar 10 2002
a(n) = d(n,0) with d(n,r) = r if n=0, otherwise d(floor(n/10), r*10+(n mod 10)). - Reinhard Zumkeller, Mar 04 2010
a(10*n+x) = x*10^m + a(n) if 10^(m-1) <= n < 10^m and 0 <= x <= 9. - Robert Israel, Jun 11 2015

Extended by Ray Chandler, Dec 30 2004

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

Original entry on oeis.org

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

A004087 Primes written backwards.

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 71, 91, 32, 92, 13, 73, 14, 34, 74, 35, 95, 16, 76, 17, 37, 97, 38, 98, 79, 101, 301, 701, 901, 311, 721, 131, 731, 931, 941, 151, 751, 361, 761, 371, 971, 181, 191, 391, 791, 991, 112, 322, 722, 922, 332, 932, 142, 152, 752, 362, 962, 172

Offset: 1

N. J. A. Sloane

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040.

a004087 n = a004087_list !! (n-1)
a004087_list = map a004086 a000040_list
-- Reinhard Zumkeller, Oct 14 2011

[Seqint(Reverse(Intseq(NthPrime(n)))): n in [1..60]]; // Vincenzo Librandi, Jan 21 2016

FromDigits[Reverse[IntegerDigits[#]]]&/@Prime[Range[100]] (* Vincenzo Librandi, Jul 05 2015 *)

from sympy import primerange
print([int(str(p)[::-1]) for p in primerange(2, 272)]) # Michael S. Branicky, Jun 24 2022

a(n) = A004086(A000040(n)) = A000040(n) - A068396(n). - N. J. A. Sloane, Jun 29 2008
a(n) = A188649(A000040(n)). - Reinhard Zumkeller, Apr 11 2011
a(n) = A071786(A000040(n)). - Reinhard Zumkeller, Oct 14 2011

More terms from Eric M. Schmidt, Apr 04 2014

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

Original entry on oeis.org

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Programs

Haskell

PARI

Python

A004086 Read n backwards (referred to as R(n) in many sequences).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

J

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Python

Sage

Formula

Extensions

A004087 Primes written backwards.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions