A002113 -id:A002113 - cp's OEIS frontend

A357044 Lexicographic earliest sequence of distinct palindromes (A002113) such that a(n)+a(n+1) is never palindromic.

1, 9, 3, 7, 5, 8, 2, 11, 4, 6, 22, 88, 44, 66, 77, 33, 99, 55, 101, 909, 111, 191, 121, 181, 131, 171, 141, 161, 151, 252, 262, 242, 272, 232, 282, 222, 292, 212, 393, 313, 494, 323, 383, 333, 373, 343, 363, 353, 454, 464, 444, 474, 434, 484, 424

Offset: 1

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

Obviously the sequence cannot contain 0.

It is easy to prove that the sequence is a permutation of the nonzero palindromes (in the sense that it contains each of them exactly once).

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. A002113 (palindromes), A029742 (non-palindromes), A262038 (next palindrome), A357045 (non-palindromes with palindromic sum of neighbors).

A357044_first(n, U=[0], a=9)={vector(n,k, k=U[1]; while(is_A002113(a+k=A262038(k+1)) || setsearch(U, k), ); U=setunion(U,[a=k]); while(#U>1 && U[2]==A262038(U[1]+1), U=U[^1]); a)}

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def nextpal(p): # next largest palindrome after palindrome p
    d = str(p)
    if set(d) == {'9'}: return int('1' + '0'*(len(d)-1) + '1')
    h = str(int(d[:(len(d)+1)//2]) + 1)
    return int(h + h[:-1][::-1]) if len(d)&1 else int(h + h[::-1])
def agen():
    aset, pal, minpal = {1}, 1, 2
    while True:
        an = pal; yield an; aset.add(an); pal = minpal
        while pal in aset or ispal(an+pal): pal = nextpal(pal)
        while minpal in aset: minpal = nextpal(minpal)
print(list(islice(agen(), 55))) # Michael S. Branicky, Sep 14 2022

A226487 First available increasing palindromes (A002113) found in the decimal expansion of the number e-2 (A001113).

Original entry on oeis.org

7, 8, 818, 2662, 9669, 39193, 94349, 99699, 985589, 988890, 5065605, 6609066, 7193917, 7390937, 8316138, 43488434, 563303365, 799929997, 1149559411, 68088588086, 85367376358, 208532235802, 991964469199

Offset: 1

Patrick De Geest and Robert G. Wilson v, Jun 09 2013

The entry 988890 is actually 0988890.

Cf. A001113, A226486.

e = RealDigits[E-2, 10, 2500000][[1]]; palQ[n_] := n == Reverse[n]; mx = 0; k = 1;   While[k < 1000000, j = 1; While[j <= k, If[ palQ[ Take[ e, {j, k}]], p = FromDigits[ Take[e, {j, k}]]; If[p > mx, mx = p; Print[p]; e = Drop[e, k]; k = 0; Break[]]]; j++]; k++]

A235922 Minimal k > 1 such that the base-k representation of the n-th palindrome in A002113, read in decimal, is also palindrome.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 10, 10, 2, 10, 4, 10, 10, 5, 2, 10, 6, 3, 10, 6, 3, 10, 7, 10, 6, 10, 3, 10, 10, 3, 5, 10, 10, 5, 7, 10, 2, 10, 8, 6, 10, 10, 4, 10, 4, 10, 8, 10, 6, 10, 10, 9, 10, 3, 10, 10, 10, 10, 10, 10, 9, 10, 10, 2, 10, 10, 10, 5, 10, 9, 3, 4, 5, 10, 10, 10, 2, 10, 10, 10, 3

Offset: 1

Vladimir Shevelev, Jan 17 2014

We conjecture that limit of average (a(1)+...+a(n))/n exists and equals 10.

Cf. A002113, A235354.

A246008 Let pal(k) denote the k-th palindrome, A002113(k), and let a(0)=0 and a(1)=1. For n >= 2, if a(n-1) = pal(k), then a(n) = pal(k+i), where i = a(n-1)-a(n-2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 33, 252, 15451, 54533545, 445335474533544, 34533547453354645335474533543, 245335474533546453354745335454533547453354645335474533542, 14533547453354645335474533545453354745335464533547453354445335474533546453354745335454533547453354645335474533541

Offset: 0

Robert G. Wilson v, Sep 20 2014

If instead we set a(n) = pal(a(n-1)), the sequence would have to start at the twelfth palindrome 22, and go from there: 12, 22, 121, 2112, 1112111, 112111111211, 1211111121111211111121, 211111121111211111121121111112111121111112, ..., . This sequence is interesting in that only the digits 1 and 2 appear so far (tested out to seventeen terms).

			For n=11, we have a(10)=11=pal(11), a(10)-a(9)=11-9=2, so a(11) = pal(11+2) = pal(13) = 33.
a(12) = 252 because it is the twenty-second palindrome after 13, the difference between 11 and 33.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..21
Hiroaki Yamanouchi, Python source code

Cf. A002113, A007097.

NextPalindrome[n_] := Block[{l = Floor[ Log[ 10, n] + 1], idn = IntegerDigits[ n]}, If[ Union[ idn] == {9}, Return[n + 2], If[l < 2, Return[n + 1], If[ FromDigits[ Reverse[ Take[ idn, Ceiling[l/2]]]] > FromDigits[ Take[ idn, -Ceiling[l/2]]], FromDigits[ Join[ Take[ idn, Ceiling[l/2]], Reverse[ Take[ idn, Floor[l/2]]]]], idfhn = FromDigits[ Take[ idn, Ceiling[l/2]]] + 1; idp = FromDigits[ Join[ IntegerDigits[ idfhn], Drop[ Reverse[ IntegerDigits[ idfhn]], Mod[l, 2]]]]]]]]; f[s_List] := Block[{a = s[[-1]], b = s[[-2]]}, Append[s, Nest[ NextPalindrome@# &, a, a - b]]]; s = {0, 1}, Nest[f, s, 14]
nthPalindrome[n_] := Block[{q = n + 1 - 10^Floor[ Log10[n + 1 - 10^Floor[ Log10[ n/10]] ]], c = Sum[ Floor[ Floor[ n/(11*10^(k - 1) - 1)]/(Floor[ n/(11*10^(k - 1) - 1)] - 1/10)] - Floor[ Floor[ n/(2*10^k - 1)]/(Floor[ n/(2*10^k - 1)] - 1/10)], {k, Floor[ Log10[ n]] }]}, Mod[q, 10]*11^c*10^Floor[ Log10[ q]] + Sum[ Floor[ Mod[q, 10^(k + 1)]/10^k]*10^(Floor[ Log10[ q]] - k) (10^(2 k + c) + 1) , {k, Floor[ Log10[ q]] }]] (* after the work of Eric A. Schmidt, see A002113 *) s = {0}; f[s_List] := Block[{k = s[[-1]] + 1}, Append[s, nthPalindrome[ k]]]; Nest[f, s, 18] (* Robert G. Wilson v, Sep 22 2014 *)

from itertools import islice
def A246008_gen(): # generator of terms
    a, b, k = 0, 1, 2
    yield 0
    while True:
        yield b
        a, b = b, int((c:=k-x)*x+int(str(c)[-2::-1] or 0) if (k:=k+b-a)<(x:=10**(len(str(k>>1))-1))+(y:=10*x) else (c:=k-y)*y+int(str(c)[::-1] or 0))
A246008_list = list(islice(A246008_gen(),22)) # Chai Wah Wu, Jul 10 2024

a(16)-a(17) from Hiroaki Yamanouchi, Sep 21 2014

Edited by N. J. A. Sloane, Oct 01 2014

A248753 Palindromes p = A002113(n) whose index n is a substring of p.

Original entry on oeis.org

11, 1111, 12221, 23332, 34443, 45554, 56665, 67776, 78887, 89998, 101101, 111111, 121121, 131131, 141141, 151151, 161161, 171171, 181181, 191191, 1020201, 1121211, 1222221, 1323231, 1424241, 1525251, 1626261, 1727271, 1828281, 1929291, 2030302, 2131312

Offset: 1

Robert G. Wilson v, Oct 13 2014

That is to say the 'n' of A002113(n) is a substring of A002113(n).

			11 is a term because the eleventh palindrome is 11.
12221 is in the sequence because the 222nd palindrome is 12221.
101101 is a member because it is the 1101st palindrome.

Cf. A002113, A248754.

(* first load 'nthPalindrome' from A002113 and then *) nPal[n_] := nthPalindrome[n - 1]; fQ[n_] := StringPosition[ ToString[ nPal[ n]], ToString[ n]] != {}; k = 1; lst = {}; While[k < 3001, If[fQ[k], AppendTo[lst, nPal[ k]]]; k++]; lst

from itertools import count, islice
def A248753_gen(startvalue=2): # generator of terms >= startvalue
    def f(n):
        y = 10*(x:=10**(len(str(n>>1))-1))
        return (c:=n-x)*x+int(str(c)[-2::-1] or 0) if nA248753_list = list(islice(A248753_gen(),32)) # Chai Wah Wu, Jul 24 2024

A248754 Palindromes p=A002113(n) whose index n is also a palindrome and in addition a substring of p (strings in base 10).

Original entry on oeis.org

11, 1111, 12221, 23332, 34443, 45554, 56665, 67776, 78887, 89998, 111111, 1222221, 2333332, 3444443, 4555554, 5666665, 6777776, 7888887, 8999998, 9101019, 11111111, 102020201, 112121211, 122222221, 132323231, 142424241, 152525251, 162626261, 172727271, 182828281

Offset: 1

Robert G. Wilson v, Oct 13 2014

This is a proper subsequence of A248753 (where the index does not need to be palindromic).

			11 is a term because the eleventh palindrome is 11.
1111 is a member because it is the 111th palindrome.
12221 is in the sequence because the 222nd palindrome is 12221.

Cf. A002113, A248753.

(* first load 'nthPalindrome' from A002113 and then *) nPal[n_] := nthPalindrome[n - 1];  fQ[n_] := StringPosition[ ToString[ nPal[ nPal[ k]]], ToString[ nPal[ n]]] != {}; k = 2; lst = {}; While[k < 501, If[ fQ[k], AppendTo[lst, nPal[ nPal[ k]] ]]; k++]; lst

from itertools import count, islice
def A248754_gen(): # generator of terms
    def f(n):
        y = 10*(x:=10**(len(str(n>>1))-1))
        return (c:=n-x)*x+int(str(c)[-2::-1] or 0) if nA248754_list = list(islice(A248754_gen(),30)) # Chai Wah Wu, Jul 24 2024

A286138 Pseudo-palindromic numbers: not palindromes (A002113), but a nontrivial palindromic concatenation (AA or ABA) of arbitrary nonzero integers A and B.

Original entry on oeis.org

1010, 1101, 1121, 1131, 1141, 1151, 1161, 1171, 1181, 1191, 1201, 1211, 1212, 1231, 1241, 1251, 1261, 1271, 1281, 1291, 1301, 1311, 1313, 1321, 1341, 1351, 1361, 1371, 1381, 1391, 1401, 1411, 1414, 1421, 1431, 1451, 1461, 1471, 1481, 1491, 1501, 1511, 1515, 1521, 1531

Offset: 1

M. F. Hasler, May 03 2017

The pseudo- or almost-palindromic numbers considered here are not related to the similarly named but different concepts mentioned in comments on A003555 and in A060087 - A060088.
We could consider "more general" palindromic concatenations like A.B.B.A, A.B.C.B.A, etc., but all of these can be written as A.B'.A with B' = B.B resp. B.C.B, etc. The result is non-palindromic (i.e., not in A002113) as required, if and only if at least one of the strings is non-palindromic.
Here, A is allowed to have only one digit, so most of the first 100 terms are of the form 1.B.1 where B = 10, 12, 13, ... (palindromes 11, 22, 33, ... excluded).
If all of the strings A, B (...) are required to be non-palindromic, the sequence starts with terms of the form A.A with A = 10, 12, 13, ..., 98:  1010, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2323, .... This is a subsequence of A239019 (numbers which are not primitive words over the alphabet {0,...,9} when written in base 10).

A286138 = select(t->!is_A002113(t),setunion(vector(801,i,((i-1)\89+1)*1001+((i-1)%89+1)*10),vector(89,i,(i+9)*101))) \\ The first 810 terms.

A302663 Lexicographically first sequence of distinct terms such that the absolute differences |a(n) - a(n+1)| are A002113(n+1), where A002113 is "the palindromes in base 10".

Original entry on oeis.org

1, 2, 4, 7, 3, 8, 14, 21, 13, 22, 11, 33, 66, 110, 55, 121, 44, 132, 231, 130, 19, 140, 9, 150, 301, 462, 291, 472, 281, 483, 271, 493, 261, 503, 251, 513, 241, 523, 815, 512, 199, 522, 189, 532, 179, 542, 169, 552, 159, 563, 149, 573, 139, 583, 129, 593, 119, 603, 109, 614, 99, 624, 89, 634, 79, 644

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Apr 11 2018

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that doesn't lead to a contradiction.

			|1 - 2| = 1, which is the 2nd palindrome of A002113 (the 1st one being "0");
|2 - 4| = 2 which is the 3rd palindrome;
|4 - 7| = 3 which is the 4th palindrome;
|7 - 3| = 4 which is the 5th palindrome;
|3 - 8| = 5 which is the 6th palindrome;
|8 - 14| = 6 which is the 7th palindrome;
|14 - 21| = 7 which is the 8th palindrome;
|21 - 13| = 8 which is the 9th palindrome;
|13 - 22| = 9 which is the 10th palindrome;
|22 - 11| = 11 which is the 11th palindrome;
|11 - 33| = 22 which is the 12th palindrome; etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..2229

Cf. A002113 (palindromes in base 10).

A319586 Number of n-digit base-10 palindromes (A002113) that cannot be written as the sum of two positive base-10 palindromes.

Original entry on oeis.org

2, 0, 8, 7, 95, 94, 975, 971, 9810, 9805, 98288, 98272

Offset: 1

Hugo Pfoertner, Sep 23 2018

			a(1) = 2, because 0 and 1 are not sums of two positive 1-digit integers, all of which are palindromes. a(3) = 8, because the 8 3-digit palindromes 111, 131, 141, 151, 161, 171, 181, and 191 (A213879(2) ... A213879(9)) cannot be written as sum of two nonzero palindromes.

Cf. A002113, A035137, A213879, A319477.

\\ calculates a(2)...a(8) using M. F. Hasler's functions in A002113
A002113(n)={my(L=logint(n,10));(n-=L=10^max(L-(n<11*10^(L-1)), 0))*L+fromdigits(Vecrev(digits(if(nA002113(n)={Vecrev(n=digits(n))==n}
inv_A002113(P)={P\(P=10^(logint(P+!P, 10)\/2))+P}
for(i=1,8,j=0;for(m=inv_A002113(10^i+1),inv_A002113(2*(10^i+1)),P=A002113(m);issum=0;for(k=2,m,PP=A002113(k);if(PP>P/2,break);if(is_A002113(P-PP),issum=1;break));if(issum==0,j++));print1(j,", ",))

from sympy import isprime
from itertools import product
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 a(n):
    palslst = [p for d in range(1, n+1) for p in pals(d)][1:]
    palsset = set(palslst)
    cs = ctot = 0
    for p in pals(n):
        ctot += 1
        for p1 in palslst:
            if p - p1 in palsset: cs += 1; break
            if p1 > p//2: break
    return ctot - cs
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Jul 12 2021

a(12) from Giovanni Resta, Oct 01 2018

A353703 Palindromes (A002113) in A157037.

Original entry on oeis.org

6, 22, 66, 202, 222, 282, 434, 454, 474, 494, 555, 595, 838, 858, 969, 1001, 1551, 1771, 3333, 3553, 5335, 6006, 6226, 6886, 8778, 9889, 12921, 14541, 15051, 16261, 16761, 17171, 18681, 19491, 20202, 20602, 20802, 20902, 24142, 24242, 24542, 28282, 28482, 30003

Offset: 1

Marius A. Burtea, May 08 2022

Intersection of A002113 and A157037.

			22 = A002113(12) and 22 = A157037(3), so 22 is a term.
66 = A002113(16) and 22 = A157037(8), so 66 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002113, A003415, A157037.

f:=func; pal:=func; [n:n in [2..30003]| pal(n) and IsPrime(Floor(f(n)))];

filter:= proc(n) local t;
  isprime(n*add(t[2]/t[1], t=ifactors(n)[2]))
end proc:
digrev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
N:= 100: # for a(1) to a(N)
Res:= 6: count:= 1:
for d from 2 while count < N do
  if d::even then
    m:= d/2;
    for n from 10^(m-1) to 10^m-1 while count < N do
      v:=  n*10^m + digrev(n);
      if filter(v) then Res:= Res,v; count:= count+1  fi;
    od
  else
    m:= (d-1)/2;
    for n from 10^(m-1) to 10^m-1 while count < N do
      for y from 0 to 9 while count < N do
        v:= n*10^(m+1)+y*10^m+digrev(n);
        if filter(v) then Res:= Res,v; count:= count+1 fi;
    od od:
  fi
od:
Res; # Robert Israel, May 09 2023

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[30003], PalindromeQ[#] && PrimeQ[d[#]] &] (* Amiram Eldar, May 09 2022 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = my(d); isprime(ad(m)) && (d=digits(m)) && (d==Vecrev(d)); \\ Michel Marcus, May 09 2022

from itertools import chain, count, islice
from sympy import isprime, factorint
def A353703_gen(): # generator of terms
    return filter(lambda n:isprime(sum(n*e//p for p,e in factorint(n).items())), chain.from_iterable(chain((int((s:=str(d))+s[-2::-1]) for d in range(10**l,10**(l+1))), (int((s:=str(d))+s[::-1]) for d in range(10**l,10**(l+1)))) for l in count(0)))
A353703_list = list(islice(A353703_gen(),20)) # Chai Wah Wu, Jun 23 2022

cp's OEIS Frontend

A357044 Lexicographic earliest sequence of distinct palindromes (A002113) such that a(n)+a(n+1) is never palindromic.

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A226487 First available increasing palindromes (A002113) found in the decimal expansion of the number e-2 (A001113).

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A235922 Minimal k > 1 such that the base-k representation of the n-th palindrome in A002113, read in decimal, is also palindrome.

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A246008 Let pal(k) denote the k-th palindrome, A002113(k), and let a(0)=0 and a(1)=1. For n >= 2, if a(n-1) = pal(k), then a(n) = pal(k+i), where i = a(n-1)-a(n-2).

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A248753 Palindromes p = A002113(n) whose index n is a substring of p.

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A248754 Palindromes p=A002113(n) whose index n is also a palindrome and in addition a substring of p (strings in base 10).

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A286138 Pseudo-palindromic numbers: not palindromes (A002113), but a nontrivial palindromic concatenation (AA or ABA) of arbitrary nonzero integers A and B.

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A302663 Lexicographically first sequence of distinct terms such that the absolute differences |a(n) - a(n+1)| are A002113(n+1), where A002113 is "the palindromes in base 10".

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A319586 Number of n-digit base-10 palindromes (A002113) that cannot be written as the sum of two positive base-10 palindromes.

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