A002113 -id:A002113 - cp's OEIS frontend

A261132 Number of ways to write n as the sum u+v+w of three palindromes (from A002113) with 0 <= u <= v <= w.

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 16, 17, 17, 18, 17, 17, 16, 15, 13, 12, 11, 10, 9, 8, 7, 7, 6, 6, 6, 6, 5, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 9, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 11, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 12, 8, 8, 8

Offset: 0

Giovanni Resta, Aug 10 2015

It is known that a(n) > 0 for every n, i.e., every number can be written as the sum of 3 palindromes.

The graph has a kind of self-similarity: looking at the first 100 values, there is a Gaussian-shaped peak centered at the first local maximum a(15) = 18. Looking at the first 10000 values, one sees just one Gaussian-shaped peak centered around the record and local maximum a(1453) = 766, but to both sides of this value there are smaller peaks, roughly at distances which are multiples of 10. In the range [1..10^6], one sees a Gaussian-shaped peak centered around the record a(164445) = 57714. In the range [1..3*10^7], there is a similar peak of height ~ 4.3*10^6 at 1.65*10^7, with smaller neighbor peaks at distances which are multiples of 10^6, etc. - M. F. Hasler, Sep 09 2018

			a(0)=1 because 0 = 0+0+0;
a(1)=1 because 1 = 0+0+1;
a(2)=2 because 2 = 0+1+1 = 0+0+2;
a(3)=3 because 3 = 1+1+1 = 0+1+2 = 0+0+3.
a(28) = 6 since 28 can be expressed in 6 ways as the sum of 3 palindromes, namely, 28 = 0+6+22 = 1+5+22 = 2+4+22 = 3+3+22 = 6+11+11 = 8+9+11.

Giovanni Resta, Table of n, a(n) for n = 0..10000
Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv: 1602.06208 [math.NT], 2016-2017.
Erich Friedman, Problem of the Month (June 1999)
James Grime and Brady Haran, Every Number is the Sum of Three Palindromes, Numberphile video (2018)
Hugo Pfoertner, Plot of first 10^6 terms
Hugo Pfoertner, Plot of first 3*10^7 terms
Hugo Pfoertner, Plot of low values in range 7*10^6 ... 10^7

Cf. A002113, A035137, A260254, A261131, A319453.

See A261422 for another version.

A261132 := proc(n)
    local xi,yi,x,y,z,a ;
    a := 0 ;
    for xi from 1 do
        x := A002113(xi) ;
        if 3*x > n then
            return a;
        end if;
        for yi from xi do
            y := A002113(yi) ;
            if x+2*y > n then
                break;
            else
                z := n-x-y ;
                if z >= y and isA002113(z) then
                    a := a+1 ;
                end if;
            end if;
        end do:
    end do:
    return a;
end proc:
seq(A261132(n),n=0..80) ; # R. J. Mathar, Sep 09 2015

pal=Select[Range[0, 1000], (d = IntegerDigits@ #; d == Reverse@ d)&]; a[n_] := Length@ IntegerPartitions[n, {3}, pal]; a /@ Range[0, 1000]

A261132(n)=n||return(1); my(c=0, i=inv_A002113(n)); A2113[i] > n && i--; until( A2113[i--]*3 < n, j = inv_A002113(D = n-A2113[i]); if( j>i, j=i, A2113[j] > D && j--); while( j >= k = inv_A002113(D - A2113[j]), A2113[k] == D - A2113[j] && c++; j--||break));c \\ For efficiency, this uses an array A2113 precomputed at least up to n. - M. F. Hasler, Sep 10 2018

a(n) = Sum_{k=0..3} A319453(n,k). - Alois P. Heinz, Sep 19 2018

Examples revised and plots for large n added by Hugo Pfoertner, Aug 11 2015

A046489 Sum of the first n palindromes (A002113).

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 56, 78, 111, 155, 210, 276, 353, 441, 540, 641, 752, 873, 1004, 1145, 1296, 1457, 1628, 1809, 2000, 2202, 2414, 2636, 2868, 3110, 3362, 3624, 3896, 4178, 4470, 4773, 5086, 5409, 5742, 6085, 6438, 6801, 7174, 7557, 7950, 8354

Offset: 0

Patrick De Geest, Sep 15 1998

Harvey P. Dale, Table of n, a(n) for n = 0..1000
P. De Geest, World!Of Numbers

Cf. A002113, A007504, A046485.

palQ[n_] := Reverse[x = IntegerDigits[n]] == x; Accumulate[Select[Range[410], palQ]] (* Jayanta Basu, Jun 26 2013 *)

A118031 Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.

Original entry on oeis.org

3, 3, 7, 0, 2, 8, 3, 2, 5, 9, 4, 9, 7, 3, 7, 3, 3, 2, 0, 4, 9, 2, 1, 5, 7, 2, 9, 8, 5, 0, 5, 5, 3, 1, 1, 2, 3, 0, 7, 1, 4, 5, 7, 7, 7, 9, 4, 5, 2, 7, 7, 8, 4, 9, 1, 3, 3, 5, 0, 6, 8, 9, 2, 5, 9, 8, 2, 5, 1, 9, 7, 6, 0, 3, 4, 9, 4, 7, 6, 7, 5, 8, 9, 7, 0, 3, 0, 1

Offset: 1

Martin Renner, May 11 2006

The sum using all palindromic numbers < 10^8 is 3.37000183240... Extrapolating using Wynn's epsilon method gives a value near 3.37018... - Eric W. Weisstein, May 14 2006

			3.3702832594973733204921572985...

Joseph Myers, Table of n, a(n) for n = 1..1001
Joseph Myers, Polynomial-time algorithm.
Michael Penn, Does this series converge??, YouTube video, 2021.
Radovan Potůček, Formulas for the Sums of the Series of Reciprocals of the Polynomial of Degree Two with Non-zero Integer Roots, Algorithms as a Basis of Modern Applied Mathematics, Studies in Fuzziness and Soft Computing book series (STUDFUZZ, Vol. 404) Springer (2021), 363-382.
Eric Weisstein's World of Mathematics, Palindromic Number.

Cf. A002113.

Similar sequences: A118064, A194097, A244162.

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]]]]]]]]; pal = 1; sm = 0; Do[ While[pal < 10^n + 1, sm = N[sm + 1/pal, 128]; pal = NextPalindrome@ pal]; Print[{n, sm}], {n, 0, 17}] (* Robert G. Wilson v, Oct 20 2010 *)

a(n) = Sum_{palindromes p>0} 1/p.

a(n) = Sum_{n>=2} 1/A002113(n).

Corrected by Eric W. Weisstein, May 14 2006
Corrected and extended by Robert G. Wilson v, Oct 20 2010
Corrected and extended by Joseph Myers, Jun 26 2014

A264406 Smallest palindrome of each distinct decimal type (A002113) in increasing order.

Original entry on oeis.org

1, 11, 101, 111, 1001, 1111, 10001, 10101, 10201, 11011, 11111, 100001, 101101, 102201, 110011, 111111, 1000001, 1001001, 1002001, 1010101, 1011101, 1012101, 1020201, 1021201, 1022201, 1023201, 1100011, 1101011, 1102011, 1110111, 1111111, 10000001, 10011001, 10022001, 10100101, 10111101, 10122101, 10200201, 10211201, 10222201, 10233201, 11000011, 11011011, 11022011, 11100111, 11111111

Offset: 1

Vladimir Shevelev, Dec 10 2015

Only positive palindromes are considered.

The numbers N(n) of distinct types of n-digit palindromes, for n=1,2,..., are 1,1,2,2,5,5,15,15,... (A164904, n>=1). It is easy to see that N(2*n-1)=N(2*n), n>=1.

			The type corresponding to the term 1021201 has the form XYZXZYX, where X,Y,Z are distinct decimal digits, X>0.

Peter J. C. Moses, Table of n, a(n) for n = 1..6450

Cf. A002113, A164904.

Two missed terms were found by Peter J. C. Moses, Jan 07 2016

A261570 Concatenation of the palindromic numbers (A002113) in increasing order up to the n-th term and then in decreasing order.

Original entry on oeis.org

1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 12345678911987654321, 123456789112211987654321, 1234567891122332211987654321, 12345678911223344332211987654321, 123456789112233445544332211987654321

Offset: 1

Robert G. Wilson v, Aug 24 2015

By definition, all terms are palindromes. Inspired by A261493.
There are no primes in this sequence up to a(1100).
The least prime factors of a(n), n>=1, are: 1, 11, 3, 11, 41, 3, 239, 11, 3, 11, 11, 3, 11, 11, 3, 11, 11, 3, 71, 21557, 19, 17, 31, 181, 17, 353, 19, 31, 19, 29, 17, 29, 11616377, 214141, 19, 5471, 17, 13883, 3, 7, ..., . See A261411.
The first (probable) prime in this sequence was found by David Broadhurst on Aug 25 2015: this is a(2007), a 21233-digit probable prime with central term 1008001. - N. J. A. Sloane, Aug 24 2015

			a(4) is the concatenation of 1, 2, 3 and 4, and then 3, 2 and 1 which results in 1234321.

M. F. Hasler, Table of n, a(n) for n = 1..108

Cf. A002113, A173426, A261493, A261411.

palQ[n_] := Reverse[idn = IntegerDigits@ n] == idn; s = Select[ Range @111, palQ]; f[n_] := FromDigits@ Flatten[ IntegerDigits@# & /@ Join[Take[s, n], Reverse@ Take[s, n - 1]]]; a = Array[f, 14]

A002113(n)=if(n>9,(n-=9)*10+if(n>9,n\10,n),n)/* This "poor man's" version is valid only for n<109 */
A261570(n,S=A002113(n))={while(n--,S=Str(A002113(n),S,A002113(n)));eval(S)} \\ M. F. Hasler, Aug 29 2015

A265641 Palindromes in base 10 (A002113) which are also prime factorization palindromes (A265640).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 44, 99, 101, 121, 131, 151, 171, 181, 191, 212, 242, 252, 272, 292, 313, 333, 343, 353, 363, 373, 383, 404, 464, 484, 575, 656, 676, 727, 747, 757, 787, 797, 828, 848, 909, 919, 929, 1331, 5445, 6336, 8228

Offset: 1

Vladimir Shevelev, Dec 11 2015

Composite numbers in the sequence have two forms of symmetry.

			5445 = 3*11*5*11*3, so it is a term.

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, n, a(n), symmetric factorization of a(n)
David A. Corneth, PARI program

Intersection of A002113 and A265640.

isok(n) = (Vecrev(m=digits(n))==m) && (isprime(core(n)) || issquare(n)); \\ Michel Marcus, Jan 15 2019

\\ See Corneth link \\ David A. Corneth, Jan 22 2019

Missing term 8 inserted by Martin Schlegel, Jan 15 2019

A282584 Number of compositions (ordered partitions) of n into decimal palindromes (A002113).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1022, 2042, 4081, 8156, 16300, 32576, 65104, 130112, 260032, 519681, 1038595, 2075660, 4148259, 8290402, 16568581, 33112734, 66176648, 132255728, 264316464, 528243231, 1055707644, 2109858797, 4216606912, 8426997041, 16841569684, 33658308890, 67266993433

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

			a(4) = 8 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes
Index entries for sequences related to compositions

Cf. A002113, A091580, A091581, A260254, A261422.

nmax = 37; CoefficientList[Series[1/(1 - Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} x^A002113(k)).

A084019 a(n) = 9's complement of n-th palindrome (A002113).

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 88, 77, 66, 55, 44, 33, 22, 11, 0, 898, 888, 878, 868, 858, 848, 838, 828, 818, 808, 797, 787, 777, 767, 757, 747, 737, 727, 717, 707, 696, 686, 676, 666, 656, 646, 636, 626, 616

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003

Leading zeros in the 9's complement are omitted.

For palindromes of the form 10^k-1 the corresponding entry is zero. Apart from this the entries are distinct.

			The 20th palindromic number A002113(20) is 101 having 9's complement 898 (999 - 101 = 898). So a(20) = 898. - _Indranil Ghosh_, Jan 30 2017

Indranil Ghosh, Table of n, a(n) for n = 1..25000

Cf. A002113, A061601, A084020.

# Program for generating the b-file
def a(n):
    return 10**len(str(n))-n-1
i=0
j=1
while j<=250:
    if i==int(str(i)[::-1]):
        print(str(j)+" "+str(a(i)))
        j+=1
    i+=1 # Indranil Ghosh, Jan 30 2017

def A084019(n):
    if n == 1: return 9
    y = 10*(x:=10**(len(str(n>>1))-1))
    if nChai Wah Wu, Jun 13 2024

A110786 To obtain a(n), take the n-th palindrome P = A002113(n) and concatenate it with the smallest palindrome Q such that PQ is a prime.

Original entry on oeis.org

11, 23, 31, 41, 53, 61, 71, 83, 97, 113, 223, 331, 443, 557, 661, 773, 881, 991, 1013, 1117, 1213, 1319, 14177, 1511, 1613, 171131, 1811, 1913, 2027, 2129, 2221, 232171, 2423, 2521, 2621, 2729, 28211, 2927, 3037, 3137, 32377, 3331, 3433, 3533

Offset: 1

Amarnath Murthy, Aug 12 2005

			The palindrome 171 gives a prime 171131 when concatenated with 131 and no palindrome less than 131 gives a prime on concatenation: 1711,1713,1717,1719,17111, etc. up to 171121 are all composite.

Cf. A030675, A110787.

from itertools import count
from sympy import isprime
def A110786(n):
    s = str((c:=n+1-x)*x+int(str(c)[-2::-1] or 0) if n+1<(x:=10**(len(str(n+1>>1))-1))+(y:=10*x) else (c:=n+1-y)*y+int(str(c)[::-1] or 0))
    for k in count(2):
        if isprime(pq:=int(s+str((c:=k-x)*x+int(str(c)[-2::-1] or 0) if k<(x:=10**(len(str(k>>1))-1))+(y:=10*x) else (c:=k-y)*y+int(str(c)[::-1] or 0)))):
            return pq # Chai Wah Wu, Jul 10 2024

More terms from Giovanni Resta, Feb 08 2006

Edited by N. J. A. Sloane, Jan 16 2009

A226486 First available increasing palindromes (A002113) found in the decimal expansion of Pi-3 (A000796).

Original entry on oeis.org

1, 4, 5, 9, 535, 979, 46264, 59195, 73637, 77477, 99999, 467764, 8683868, 23911932, 398989893, 559555955, 769646967, 972464279, 992868299, 21348884312, 49612121694, 450197791054, 9475082805749

Offset: 1

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

Cf. A000796, A068046, A226487.

pi = RealDigits[Pi-3, 10, 2500000][[1]]; palQ[n_] := n == Reverse[n]; mx = 0; k = 1; While[k < 1000, j = 1; While[j <= k, If[ palQ[ Take[ pi, {j, k}]], p = FromDigits[ Take[ pi, {j, k}]]; If[p > mx, mx = p; Print[p]; pi = Drop[pi, k]; k = 0; Break[]]]; j++]; k++]

cp's OEIS Frontend

A261132 Number of ways to write n as the sum u+v+w of three palindromes (from A002113) with 0 <= u <= v <= w.

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A046489 Sum of the first n palindromes (A002113).

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A118031 Decimal expansion of the sum of the reciprocals of the palindromic numbers A002113.

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A264406 Smallest palindrome of each distinct decimal type (A002113) in increasing order.

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A261570 Concatenation of the palindromic numbers (A002113) in increasing order up to the n-th term and then in decreasing order.

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A265641 Palindromes in base 10 (A002113) which are also prime factorization palindromes (A265640).

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PARI

PARI

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A282584 Number of compositions (ordered partitions) of n into decimal palindromes (A002113).

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Mathematica

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A084019 a(n) = 9's complement of n-th palindrome (A002113).

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