A035137 -id:A035137 - cp's OEIS frontend

A261918 Numbers which in base 5 are neither palindromes nor the sum of two palindromes.

11, 17, 23, 51, 131, 141, 146, 147, 149, 151, 153, 154, 163, 164, 169, 173, 175, 177, 179, 181, 184, 185, 194, 195, 199, 200, 201, 203, 205, 206, 211, 215, 221, 225, 226, 229, 231, 236, 237, 241, 251, 259, 261, 262, 263, 266, 267, 271, 281, 287, 289, 291, 296, 297

Offset: 1

N. J. A. Sloane, Sep 13 2015

The terms less than 20000, and conjecturally all terms, are the sum of three base 5 palindromes.

N. J. A. Sloane, Table of n, a(n) for n = 1..6235

Cf. A029952, A261917; A035137 (base 10 analog).

A115336 a(n) is the smallest number representable in exactly n ways as a sum of 2 palindromes (each of them >= 0).

Original entry on oeis.org

0, 2, 4, 6, 8, 110, 353, 363, 373, 383, 393, 464, 474, 504, 484, 494, 575, 605, 585, 1049, 595, 767, 706, 686, 777, 696, 807, 787, 878, 13222, 797, 908, 888, 31812, 12892, 898, 989, 11220, 44444, 1201, 999, 28882, 11110, 42623, 30092, 1100, 11000, 36153

Offset: 1

Giovanni Resta, Jan 20 2006

			a(6)=110 since 110=101+9=99+11=88+22=77+33=66+44=55+55 and no
number less than 110 has 6 such decompositions.

Cf. A035137, A115337.

palQ[n_] := n == FromDigits@Reverse@IntegerDigits@n; pt = Select[Range[0, 50005], palQ]; t = Array[0&, 50000]; Do[v = pt[[i]]+pt[[j]]; If[v<50000, t[[v + 1]]++ ], {i, 600}, {j, i}]; Table[Position[t, k][[1, 1]]-1, {k, 55}]

A262087 Largest palindrome p such that n-p is again a palindrome, or 0 if no such p exists.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 0, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 0, 33, 33, 33, 33, 33, 33, 33, 33, 33, 33, 0, 44, 44, 44, 44, 44, 44, 44, 44, 44, 44, 0, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 0, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 0, 77, 77, 77, 77, 77, 77

Offset: 0

M. F. Hasler, Sep 10 2015

Markus Sigg, Table of n, a(n) for n = 0..10000

Cf. A261423, A002113, A035137.

a(N,P=A261423(N))=until(N<2*P=A261423(P-1),is_A002113(N-P)&&return(P))

a(n) = 0 for n in {0} union A035137.

A282585 Number of ways to write n as an ordered sum of 3 squarefree palindromes (A071251).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 7, 9, 12, 19, 21, 21, 18, 24, 27, 28, 18, 18, 19, 24, 15, 10, 6, 12, 12, 12, 9, 9, 12, 15, 18, 12, 9, 7, 15, 15, 15, 9, 12, 15, 18, 18, 12, 9, 9, 18, 15, 12, 0, 9, 9, 9, 0, 0, 0, 6, 6, 9, 12, 9, 12, 15, 18, 18, 12, 9, 13, 18, 18, 18, 9, 15, 18, 21, 18, 12, 9, 15, 21, 21, 21, 9, 18, 21, 24, 18

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

Every number can be written as the sum of 3 palindromes (see A261132 and A261422).
Conjecture: a(n) > 0 for any sufficiently large n.
Additional conjecture: every number > 3 can be written as the sum of 4 squarefree palindromes.

			a(22) = 6 because we have [11, 6, 5], [11, 5, 6] [6, 11, 5], [6, 5, 11], [5, 11, 6] and [5, 6, 11].

Ilya Gutkovskiy, Extended graphical example
Ilya Gutkovskiy, Extended graphical example for additional conjecture
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to palindromes

Cf. A002113, A005117, A035137, A071251, A091580, A091581, A260254, A261131, A261132, A261422, A280210, A282584.

nmax = 85; CoefficientList[Series[Sum[Boole[SquareFreeQ[k] && PalindromeQ[k]] x^k, {k, 1, nmax}]^3, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} x^A071251(k))^3.

A261909 A261908(n)/9.

Original entry on oeis.org

1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 110, 111, 121, 131, 141, 151, 161, 171, 181, 191, 192, 200, 201, 212, 220, 231, 232, 252, 262, 272, 282, 292, 293, 300, 302, 312, 313, 321

Offset: 1

N. J. A. Sloane, Sep 09 2015

Created in an attempt to understand A035137. It would be nice to have an independent definition of these numbers.

Cf. A261908, A035137.

A262528 Maximum number of backward steps k needed to find a representation of an n-digit decimal number x as a sum of three base-10 palindromes of the form k-th largest base-10 palindrome <= x plus a number representable as sum of two base-10 palindromes from A260255.

Original entry on oeis.org

0, 1, 1, 3, 3, 11, 4, 10, 4, 23, 9, 15, 6, 23, 11

Offset: 1

Hugo Pfoertner, Sep 26 2015

The sequence terms are counterexamples to the second part of the claim stated in the answers to the Math Magic Problem of the Month (June 1999) that "all sufficiently large numbers seem to be the sum of 3 palindromes, one of which is the biggest or second biggest possible", which would mean all a(k)=2 for k "sufficiently large".
Since exhaustive search is currently (2015) considered as not feasible, a(16)>=16, a(17)>=7, a(18)>=25, a(19)>=14 are only lower bounds for the next sequence terms.
M. Sigg has shown that a(n)>=3 for n = 5 + 4 * j.

			a(1)=0 because all 1-digit numbers are palindromes,
a(2)=a(3)=1 because all 2-digit and all 3-digit numbers can be represented by the nearest smaller palindrome and a number <=10, e.g., 201=191+9+1.
a(4)=3, because for the number 2023 the largest palindrome leading to a difference representable as sum of two palindromes is 1881. 2023-2002=21 and 2023-1991=32 are not in A260255. 2023-1881=142=141+1 is in A260255. No other 4-digit number requires more than 3 backward steps.
a(6)=11 because for the 6-digit number 101199 none of the first 10 differences 101199-101101=98, 101199-10001=1198, 101199-99999=1200, 101199-99899=1300, 101199-99799=1400, 101199-99699=1500, 101199-99599=1600, 101199-99499=1700, 101199-99399=1800, 101199-99299=1900 is representable as sum of two palindromes (i.e., are in A035137), whereas the 11th palindrome 99199 leads to 101199-99199=2000=1991+9.
a(18)>=25 because for the number x=100000001814566071 only the 25th palindrome < x 99999997779999999 produces the first difference 4034566072 representable as sum of 2 palindromes.

Erich Friedman, Problem of the Month (June 1999)
Markus Sigg, On a conjecture of John Hoffman regarding sums of palindromic numbers, arXiv:1510.07507 [math.NT], 2015.

Cf. A002113, A035137, A088601, A109326, A260255, A261132, A261675.

A263994 First element of first run of at least n consecutive numbers not representable as the sum of two base-10 palindromes.

Original entry on oeis.org

21, 1041, 1051, 1061, 1071, 1081, 1091, 107209, 108429, 109803, 10715097, 10854691, 10904639, 10904731, 10904731, 10904731, 10904731, 10983831, 11002311, 11002311, 11002311, 1078112970, 1078122970

Offset: 1

Hugo Pfoertner, Oct 31 2015

First occurrence of n consecutive numbers in A035137.

			a(2)=1041 because 1041 and 1042 are the first two consecutive numbers in A035137.

Hugo Pfoertner, Table of n, a(n) for n = 1..52. Checked up to 2*10^11.

Cf. A002113, A035137, A260255.

A287961 Numbers that are the sum of two palindromic primes (A002385).

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 22, 103, 104, 106, 108, 112, 133, 134, 136, 138, 142, 153, 154, 156, 158, 162, 183, 184, 186, 188, 192, 193, 194, 196, 198, 202, 232, 252, 262, 282, 292, 302, 312, 315, 316, 318, 320, 322, 324, 332, 342, 355, 356, 358, 360, 362, 364, 372, 375, 376, 378, 380

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

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

Cf. A002113, A002385, A014091, A035137, A260254, A260255, A261906.

nmax = 380; f[x_] := Sum[Boole[PalindromeQ[k] && PrimeQ[k]] x^k, {k, 1, nmax}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A319387 Smallest palindrome p such that n-p is again a palindrome, or n if no such p exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 32, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 43, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 54, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 65, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 76, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 87, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 98, 0

Offset: 0

Markus Sigg, Sep 18 2018

a(n) = n if and only if A262087(n) = 0.

			a(11) = 0 because 11 = 11 + 0, so 0 is the smallest palindrome in any partitioning of 11 as a sum of two palindromes.
a(21) = 21 because 21 cannot be written as a sum of two palindromes.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A002113, A035137, A260254, A262087.

isP := k -> StringTools[IsPalindrome](convert(k,string)):
a := NULL:
for n from 0 to 99 do
   an := n:
   for k from 0 to n/2 do
      if isP(k) and isP(n-k) then an := min(an,k) end if
   end do:
   a := a,an
end do:
a;

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

cp's OEIS Frontend

A261918 Numbers which in base 5 are neither palindromes nor the sum of two palindromes.

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A115336 a(n) is the smallest number representable in exactly n ways as a sum of 2 palindromes (each of them >= 0).

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Mathematica

A262087 Largest palindrome p such that n-p is again a palindrome, or 0 if no such p exists.

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A282585 Number of ways to write n as an ordered sum of 3 squarefree palindromes (A071251).

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A261909 A261908(n)/9.

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A262528 Maximum number of backward steps k needed to find a representation of an n-digit decimal number x as a sum of three base-10 palindromes of the form k-th largest base-10 palindrome <= x plus a number representable as sum of two base-10 palindromes from A260255.

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A263994 First element of first run of at least n consecutive numbers not representable as the sum of two base-10 palindromes.

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A287961 Numbers that are the sum of two palindromic primes (A002385).

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Mathematica

A319387 Smallest palindrome p such that n-p is again a palindrome, or n if no such p exists.

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Maple

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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