A002113 -id:A002113 - cp's OEIS frontend

A357045 Lexicographically earliest sequence of distinct non-palindromic numbers (A029742) such that a(n)+a(n+1) is always a palindrome (A002113).

10, 12, 21, 23, 32, 34, 43, 45, 54, 47, 19, 14, 30, 25, 41, 36, 52, 49, 17, 16, 28, 27, 39, 38, 50, 51, 15, 18, 26, 29, 37, 40, 48, 53, 13, 20, 24, 31, 35, 42, 46, 65, 56, 75, 76, 85, 86, 95, 96, 106, 116, 126, 136, 146, 157, 105, 97, 64, 57, 74

Offset: 1

Eric Angelini and M. F. Hasler, Sep 14 2022

Conjecture: The sequence contains all non-palindromic numbers (A029742).

Eric Angelini, Sums with palindromes, personal blog "Cinquante signes" on blogspot.com, and post to the math-fun list, Sep 12 2022
Eric Angelini, Sums with palindromes, personal blog "Cinquante signes" on blogspot.com, and post to the math-fun list, Sep 12 2022 [Cached copy]

Cf. A029742 (non-palindromes), A002113 (palindromes), A357044 (palindromes with non-palindromic sum of neighbors).

A357045_first(n, U=[9], a=1)={vector(n, k, k=U[1]; until( is_A002113(a+k) && !is_A002113(k) && !setsearch(U, k), k++); U=setunion(U,[a=k]); while(#U>1 && U[2]==U[1]+1+is_A002113(U[1]+1), U=U[^1]); a)}

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def agen():
    aset, k, mink = {10}, 10, 12
    while True:
        an = k; yield an; aset.add(an); k = mink
        while k in aset or ispal(k) or not ispal(an+k): k += 1
        while mink in aset: mink += 1
print(list(islice(agen(), 60))) # Michael S. Branicky, Sep 14 2022

A071278 Sorted list of numbers b(i)*b(j), 2 < i < j, where b = A002113 = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33 ... is a list of the decimal palindromes.

Original entry on oeis.org

6, 8, 10, 12, 12, 14, 15, 16, 18, 18, 20, 21, 22, 24, 24, 27, 28, 30, 32, 33, 35, 36, 40, 42, 44, 44, 45, 48, 54, 55, 56, 63, 66, 66, 66, 72, 77, 88, 88, 88, 99, 99, 110, 110, 132, 132, 132, 132, 154, 154, 165, 165, 176, 176, 176, 198, 198, 198, 198, 202, 220, 220, 222

Offset: 1

Amarnath Murthy, Jun 07 2002

Old name was: Palindromes which are nontrivial multiples of 2 distinct palindromes.

Repetitions are allowed. 12 appears twice because it is both 2*6 and 3*4.

			8 is in the sequence because 8=2*4.

Cf. A002113.

Module[{nn=250,pals},pals=Select[Range[2,nn],PalindromeQ];Select[Sort[ Times@@#&/@ Subsets[pals,{2}]],#<=nn&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 27 2018 *)

Corrected and extended by Sascha Kurz, Jan 02 2003

Edited (with corrected definition) by N. J. A. Sloane, May 27 2018

A099994 Bisection of A002113.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 33, 55, 77, 99, 111, 131, 151, 171, 191, 212, 232, 252, 272, 292, 313, 333, 353, 373, 393, 414, 434, 454, 474, 494, 515, 535, 555, 575, 595, 616, 636, 656, 676, 696, 717, 737, 757, 777, 797, 818, 838, 858, 878, 898, 919, 939, 959, 979, 999, 1111, 1331, 1551, 1771, 1991

Offset: 1

N. J. A. Sloane, Nov 20 2004

r[n_]:=FromDigits[Reverse[IntegerDigits[n]]];v={}; Do[If[r[n]==n, v=Append[v, n]], {n, 0, 2200}];Table[v[[2k-1]], {k, Floor[Length[v]]/2}]

Extended with Mathematica program by Farideh Firoozbakht, Dec 01 2004

A099995 Bisection of A002113.

Original entry on oeis.org

1, 3, 5, 7, 9, 22, 44, 66, 88, 101, 121, 141, 161, 181, 202, 222, 242, 262, 282, 303, 323, 343, 363, 383, 404, 424, 444, 464, 484, 505, 525, 545, 565, 585, 606, 626, 646, 666, 686, 707, 727, 747, 767, 787, 808, 828, 848, 868, 888, 909, 929, 949, 969, 989, 1001, 1221, 1441, 1661, 1881, 2002

Offset: 1

N. J. A. Sloane, Nov 20 2004

r[n_]:=FromDigits[Reverse[IntegerDigits[n]]];v={}; Do[If[r[n]==n, v=Append[v, n]], {n, 0, 2200}];Table[v[[2k]], {k, Floor[Length[v]]/2}]

Extended with Mathematica program by Farideh Firoozbakht, Dec 01 2004

A101034 Numbers n such that A002113(n) is a triangular number.

Original entry on oeis.org

0, 1, 3, 6, 14, 15, 26, 68, 75, 129, 158, 186, 249, 759, 1616, 1827, 2268, 2679, 4543, 6072, 6675, 7294, 13512, 16146, 27871, 112640, 116339, 152889, 161727, 239533, 260487, 404161, 670038, 685744, 767718, 973504, 2313206, 6250177, 6977617

Offset: 1

Klaus Brockhaus, Nov 27 2004

Indices of triangular numbers in the sequence of palindromes.

			A002113(26) = 171 is a triangular number, so 26 is a term.

Cf. A000217, A002113, A003098, A008509.

var c,n,m:integer; end; begin c:=0; for n:=0 to 100000000 do if n = intreverse(n) then m:=floor(sqrt(2*n)); if m*(m+1) div 2 = n then write(c,","); end; inc(c); end; end; end;

Join[{0},Flatten[Position[Select[Range[10^7],PalindromeQ],?(OddQ[Sqrt[ 8#+1]]&)]]] (* The program generates the first 22 terms of the sequence. *) (* _Harvey P. Dale, Jun 17 2022 *)

{c=0;for(n=0,10000000,k=n;r=0;while(k>0,d=divrem(k,10);k=d[1];r=10*r+d[2]); if(n==r,m=sqrtint(2*n);if(m*(m+1)/2==n,print1(c,","));c++))}

a(24) to a(38) from Klaus Brockhaus, Oct 05 2005

A110057 Number of solutions to x == 0 (mod A007954(x)), x in A002113, 10^(n-1) <= x < 10^n.

Original entry on oeis.org

9, 1, 2, 3, 7, 3, 6, 4, 9, 9, 18, 11, 24, 17, 25, 12, 45, 16, 53, 22

Offset: 1

Jean-Christophe Colin (colinjeanchristophe(AT)yahoo.fr), Sep 04 2005

Carlos Rivera, Puzzle 41. Palindromic Carnival, The Prime Puzzles and Problems Connection.

Cf. A002113, A007954, A117057.

Edited by R. J. Mathar, Feb 08 2008

Offset changed to 1 and a(1) inserted by Jinyuan Wang, Jun 15 2022

A113614 Least number whose absolute difference of successive digits gives the n-th palindrome (A002113), or 0 if no such number exists.

Original entry on oeis.org

121, 131, 141, 151, 161, 171, 181, 191, 0, 1001, 1012, 1021, 1032, 1043, 1054, 1065, 1076, 1087, 1098, 0

Offset: 10

Amarnath Murthy, Nov 09 2005

a(n) = 0 if n begins with 9 and contains odd number of digit. E.g., 919, 92129, etc. Are there other values of n for which a(n) = 0?

If n begins with 9 as a digit and contains even number of digits then a(n) is a palindrome.

			a(99) = 909.
a(191) = 1098. 1 - 0 = 1, abs(0 - 9) = 9, 9 - 8 = 1.
a(9119) = 90109.

Cf. A002113.

A346919 Numbers that are both palindromes (A002113) and terms of A072389.

Original entry on oeis.org

0, 4, 252, 2112, 2772, 6336, 21012, 27072, 42924, 48384, 48984, 63036, 252252, 297792, 407704, 2327232, 2572752, 2747472, 2774772, 2958592, 4457544, 4811184, 6378736, 6396936, 25777752, 27633672, 29344392, 63099036, 63399336, 404080404, 409757904, 441525144

Offset: 1

Dumitru Damian, Aug 07 2021

Cf. A002113, A072389, A085780.

p[n_] :=  n*(n + 1); m = 1000; Select[Union[ Times @@@ Tuples[p /@ Range[0, m], {2}]], # <= 2*p[m] && PalindromeQ[#] &] (* Amiram Eldar, Aug 13 2021 *)

# the sequence numbers up to a limit
def a346919_upto(lim, alst=set([0])):
    for i in range(1, int(lim**0.25)):
        for j in range(i, int((lim/(i*(i+1)))**0.5)+1):
            if all([(k:=i*(i+1)*j*(j+1))<=lim, str(k)==str(k)[::-1]]): alst.add(k)
    return sorted(alst)
print(a346919_upto(10**9)) # Dumitru Damian, Apr 05 2023

Intersection of A072389 and A002113.

Intersection of 4*A085780 and A002113.

More terms from Jinyuan Wang, Aug 07 2021

A353217 Triangular numbers (A000217) with arithmetic derivative (A003415) a palindrome (A002113).

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 136, 153, 231, 741, 1711, 11026, 22366, 99681, 104653, 593505, 1348903, 1378630, 1886653, 3098805, 4388203, 4474536, 24587578, 26626753, 32092066, 45825951, 132804253, 165283471, 197239591, 355657785, 498727153, 866008153, 1074091726, 1144165366

Offset: 1

Marius A. Burtea, Apr 30 2022

			15 = A000217(5) and 15' = 8 = A002113(9), so 15 is a term.
153 = A000217(17) and 153' = 111 = A002113(21), so 153 is a term.

Cf. A000217, A003415, A002113, A068312.

tr:=func; pal:=func; f:=func; [n:n in [d*(d+1) div 2:d in [0..150000]]| pal(Floor(f(n)))];

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Table[n*(n + 1)/2, {n, 0, 50000}], PalindromeQ[d[#]] &] (* Amiram Eldar, Apr 30 2022 *)

from itertools import chain, count, islice
from sympy import factorint
def A353217_gen(): # generator of terms
    return chain((0,1),filter(lambda m:(s:=str(sum((m*e//p for p,e in factorint(m).items()))))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],(n*(n+1)//2 for n in count(2))))
A353217_list = list(islice(A353217_gen(),20)) # Chai Wah Wu, Jun 24 2022

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

cp's OEIS Frontend

A357045 Lexicographically earliest sequence of distinct non-palindromic numbers (A029742) such that a(n)+a(n+1) is always a palindrome (A002113).

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A071278 Sorted list of numbers b(i)*b(j), 2 < i < j, where b = A002113 = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33 ... is a list of the decimal palindromes.

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A099994 Bisection of A002113.

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A099995 Bisection of A002113.

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A101034 Numbers n such that A002113(n) is a triangular number.

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A110057 Number of solutions to x == 0 (mod A007954(x)), x in A002113, 10^(n-1) <= x < 10^n.

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A113614 Least number whose absolute difference of successive digits gives the n-th palindrome (A002113), or 0 if no such number exists.

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A346919 Numbers that are both palindromes (A002113) and terms of A072389.

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A353217 Triangular numbers (A000217) with arithmetic derivative (A003415) a palindrome (A002113).

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