A136522 -id:A136522 - cp's OEIS frontend

A341191 Number of ways to write n as an ordered sum of 2 nonzero decimal palindromes.

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

Offset: 2

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 2..20000

Cf. A002113, A136522, A260254, A319468, A341155, A341192, A341193.

nmax = 90; CoefficientList[Series[Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]^2, {x, 0, nmax}], x] // Drop[#, 2] &

A341192 Number of ways to write n as an ordered sum of 3 nonzero decimal palindromes.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 52, 60, 66, 70, 72, 72, 70, 66, 60, 55, 45, 39, 34, 30, 27, 25, 24, 24, 24, 27, 27, 25, 27, 27, 27, 27, 27, 27, 27, 27, 30, 27, 27, 30, 30, 30, 30, 30, 30, 30, 30, 33, 27, 30, 33, 33, 33, 33, 33, 33, 33, 33, 36, 27, 34, 36, 36, 36, 36

Offset: 3

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 3..20000

Cf. A002113, A136522, A261131, A261132, A261422, A341156, A341191, A341193.

nmax = 70; CoefficientList[Series[Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]^3, {x, 0, nmax}], x] // Drop[#, 3] &

A341193 Number of ways to write n as an ordered sum of 4 nonzero decimal palindromes.

Original entry on oeis.org

1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 274, 336, 399, 460, 516, 564, 601, 624, 636, 628, 609, 580, 544, 504, 463, 424, 390, 360, 345, 332, 324, 324, 327, 332, 338, 344, 350, 352, 356, 364, 361, 364, 370, 376, 382, 388, 394, 400, 402, 412, 418, 412, 418, 424, 430, 436, 442

Offset: 4

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 4..20000

Cf. A002113, A136522, A319469, A341157, A341191, A341192.

nmax = 60; CoefficientList[Series[Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

A047813 Largest palindromic substring of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 22, 3, 4, 5, 6, 7, 8, 9, 3, 3, 3, 33, 4, 5, 6, 7, 8, 9, 4, 4, 4, 4, 44, 5, 6, 7, 8, 9, 5, 5, 5, 5, 5, 55, 6, 7, 8, 9, 6, 6, 6, 6, 6, 6, 66, 7, 8, 9, 7, 7, 7, 7, 7, 7, 7, 77, 8, 9, 8, 8, 8, 8, 8, 8, 8, 8, 88, 9, 9, 9, 9, 9, 9, 9

Offset: 0

N. J. A. Sloane

a(n) = A262188(n,A262190(n)-1). - Reinhard Zumkeller, Sep 14 2015

			a(1313) = Max{1,3,131,313} = 313.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to palindromes

Cf. A136522.

Cf. A262188, A262223, A262224.

a047813 = last . a262188_row
-- Reinhard Zumkeller, Sep 14 2015, Aug 23 2011

palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[ idn]]; f[n_] := Block[{id = IntegerDigits@ n, mx = -Infinity}, k = Length@ id; While[k > 0 && mx == -Infinity, mx = Max[mx, Select[ FromDigits@# & /@ Partition[id, k, 1], palQ[#, 10] &]]; k--]; mx] (* Robert G. Wilson v, Aug 24 2011 *)
lps[n_]:=Module[{idn=IntegerDigits[n]},Max[FromDigits/@Select[ Flatten[ Table[ Partition[ idn,i,1],{i,Length[idn]}],1],#==Reverse[#]&]]]; Array[ lps,100,0] (* Harvey P. Dale, Jan 09 2015 *)

def c(s): return (s[0] != "0" or s == "0") and s == s[::-1]
def a(n):
    s = str(n)
    ss = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1))
    return max(int(w) for w in ss if c(w))
print([a(n) for n in range(96)]) # Michael S. Branicky, Sep 18 2022

A062293 Smallest multiple k*n of n which has even digits and is a palindrome or becomes a palindrome when 0's are added on the left (e.g., 10 becomes 010, which is a palindrome).

Original entry on oeis.org

0, 2, 2, 6, 4, 20, 6, 686, 8, 666, 20, 22, 60, 2002, 686, 60, 80, 646, 666, 646, 20, 6006, 22, 828, 600, 200, 2002, 8886888, 868, 464, 60, 868, 800, 66, 646, 6860, 828, 222, 646, 6006, 40, 22222, 6006, 68886, 44, 6660, 828, 282, 4224, 686, 200, 42024, 4004, 424, 8886888, 220, 8008, 68286, 464, 68086, 60

Offset: 0

Amarnath Murthy, Jun 18 2001

Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29 2001

			a(7) = 686 as 686 = 98*7 is the smallest palindrome multiple of 7 with even digits.

Cf. A062279. Values of k are given in A061797.

Cf. A014263, A136522, A004151.

stop := 500000; for n := 0 to 60 do k := 1; test := true; while test and k < stop do m := omit_trailzeros(n*k); if test := not all_even(m) or m <> int_reverse(m) then inc(k); end; end; if k < stop then write(n*k," "); else write(-1," "); end; end;

a062293 0 = 0
a062293 n = head [x | x <- map (* n) [1..],
                 all (`elem` "02468") $ show x, a136522 (a004151 x) == 1]
-- Reinhard Zumkeller, Feb 01 2012

Corrected and extended by Klaus Brockhaus, Jun 21 2001

A015976 One iteration of Reverse and Add is needed to reach a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 43, 44, 45, 47, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 65, 70, 71, 72, 74, 80, 81, 83, 90, 92, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Robert G. Wilson v

The number of iterations starts at 1, so palindromes (cf. A002113) are not excluded. The corresponding sequence excluding palindromes is A065206.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Reverse and Add!

Cf. A002113, A065206.

a015976 n = a015976_list !! (n-1)
a015976_list = filter ((== 1) . a136522 . a056964) [1..]
-- Reinhard Zumkeller, Oct 14 2011

rev[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Select[Range[106], rev[x = # + rev[#]] == x &] (* Jayanta Basu, Jul 24 2013 *)
Select[Range[120],PalindromeQ[#+IntegerReverse[#]]&] (* Harvey P. Dale, Jul 04 2022 *)

Offset corrected by Reinhard Zumkeller, Oct 14 2011

A078715 Palindromic Roman numerals.

Original entry on oeis.org

1, 2, 3, 5, 10, 19, 20, 30, 50, 100, 190, 200, 300, 500, 1000, 1900, 2000, 3000

Offset: 1

Rick L. Shepherd, Dec 19 2002

This sequence is consistent with the Roman numerals as expressed in the Schildberger link. 4 (usually IV now) could be included in a variant of this sequence as IIII is sometimes used (especially on clock faces). To make this or similar sequences well-defined for numbers larger than 3999, it must be decided whether and how to handle the apostrophus (backward-C), the vinculum (bar), the frame, or even other multiplier notations used at various times in representations of larger Roman numerals. Recalling the "Y2K crisis", will there be a(n even milder) "Y4M crisis"? In particular, is 4000 to be represented as MMMM, (I)(I)(I)(I) (where parentheses are used to represent C and the apostrophus), MV (with vinculum over the V), IV (with vinculum over both I and V) or IIII with vinculum over all four I's? If there is no general agreement, could Roman civilization be at risk (once again)?

Indices of terms in A061493 which are also in A002113. - M. F. Hasler, Jan 12 2015

			I, II, III, V, X, XIX, XX, XXX, L, C, CXC, CC, CCC, D, M, MCM, MM, MMM

Encyclopaedia Britannica, 1981 ed., Vol. 11, "Mathematics, History of", p. 647.
Webster's Third New International Dictionary (Unabridged), 1976 ed., "Cardinal Numbers Table" and footnotes, p. 1549.

Paul Lewis, Roman numerals and calendar
Gerard Schildberger, The first 3999 numbers in Roman numerals
Eric Weisstein's World of Mathematics, Roman Numerals

Cf. A061493, A006968 (Roman numerals main entry), A002113 (Palindromic Arabic numerals).

Subsequence of A093703.

a078715 n = a078715_list !! (n-1)
a078715_list = filter ((== 1) . a136522 . a061493) [1..3999]
-- Reinhard Zumkeller, Apr 14 2013

Select[Range[3000], PalindromeQ[RomanNumeral[#]] &] (* Paolo Xausa, Mar 03 2024 *)

is_A078715(n)=Vecrev(n=Str(A061493(n)))==Vec(n) \\ M. F. Hasler, Jan 12 2015

A136522(A061493(a(n))) = 1. - Reinhard Zumkeller, Apr 14 2013

A030147 Palindromes in which parity of digits alternates.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 121, 141, 161, 181, 212, 232, 252, 272, 292, 303, 323, 343, 363, 383, 414, 434, 454, 474, 494, 505, 525, 545, 565, 585, 616, 636, 656, 676, 696, 707, 727, 747, 767, 787, 818, 838, 858, 878, 898, 909, 929, 949

Offset: 1

Patrick De Geest

All terms have an odd number of digits. - Alonso del Arte, Jan 31 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to palindromes

Intersection of A002113 and A030141.
Subsequence of A001633.
Cf. A030143, A030144, A030152.

a030147 n = a030147_list !! (n-1)
a030147_list = filter ((== 1) . a228710) a002113_list
-- Reinhard Zumkeller, Aug 31 2013

palQ[n_, b_:10] := (IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]); alternParQ[n_, b_:10] := (Union[BlockMap[Xor @@ # &, OddQ[IntegerDigits[n, b]], 2, 1]] == {True}); Join[Range[0, 9], Select[Range[1000], palQ[#] && alternParQ[#] &]] (* Alonso del Arte, Feb 02 2020 *)
Join[Range[0,9],Select[Range[100000],PalindromeQ[#]&&Union[Total/@Partition[Boole[ EvenQ[ IntegerDigits[ #]]],2,1]] =={1}&]] (* Harvey P. Dale, Jul 04 2023 *)

def isPal(n: Int) = (n.toString == n.toString.reverse)
def alternsPar(n: Int): Boolean = {
  val dPars = Integer.toString(n).toList.map(_ % 2 == 0)
  val scanPars = (dPars zip dPars.tail).map{ case (x, y) => x ^ y }
  scanPars.toSet == Set(true)
}
(0 to 9) ++: (10 to 999).filter(isPal).filter(alternsPar) // Alonso del Arte, Feb 02 2020

A136522(a(n)) * A228710(a(n)) = 1. - Reinhard Zumkeller, Aug 31 2013

A262188 Table read by rows: row n contains all distinct palindromes contained as substrings in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Sep 14 2015

Length of row n = A262190(n);
T(n,0) = A054054(n);
T(n,A262190(n)-1) = A047813(n).

			.     n |  T(n,*)           n |  T(n,*)              n |  T(n,*)
.  -----+-----------    ------+-------------    -------+--------------
.   100 |  0,1           1000 |  0,1             10000 |  0,1
.   101 |  0,1,101       1001 |  0,1,1001        10001 |  0,1,10001
.   102 |  0,1,2         1002 |  0,1,2           10002 |  0,1,2
.   103 |  0,1,3         1003 |  0,1,3           10003 |  0,1,3
.   104 |  0,1,4         1004 |  0,1,4           10004 |  0,1,4
.   105 |  0,1,5         1005 |  0,1,5           10005 |  0,1,5
.   106 |  0,1,6         1006 |  0,1,6           10006 |  0,1,6
.   107 |  0,1,7         1007 |  0,1,7           10007 |  0,1,7
.   108 |  0,1,8         1008 |  0,1,8           10008 |  0,1,8
.   109 |  0,1,9         1009 |  0,1,9           10009 |  0,1,9
.   110 |  0,1,11        1010 |  0,1,101         10010 |  0,1,1001
.   111 |  1,11,111      1011 |  0,1,11,101      10011 |  0,1,11,1001
.   112 |  1,2,11        1012 |  0,1,2,101       10012 |  0,1,2,1001
.   113 |  1,3,11        1013 |  0,1,3,101       10013 |  0,1,3,1001
.   114 |  1,4,11        1014 |  0,1,4,101       10014 |  0,1,4,1001
.   115 |  1,5,11        1015 |  0,1,5,101       10015 |  0,1,5,1001
.   116 |  1,6,11        1016 |  0,1,6,101       10016 |  0,1,6,1001
.   117 |  1,7,11        1017 |  0,1,7,101       10017 |  0,1,7,1001
.   118 |  1,8,11        1018 |  0,1,8,101       10018 |  0,1,8,1001
.   119 |  1,9,11        1019 |  0,1,9,101       10019 |  0,1,9,1001
.   120 |  0,1,2         1020 |  0,1,2           10020 |  0,1,2
.   121 |  1,2,121       1021 |  0,1,2           10021 |  0,1,2
.   122 |  1,2,22        1022 |  0,1,2,22        10022 |  0,1,2,22
.   123 |  1,2,3         1023 |  0,1,2,3         10023 |  0,1,2,3
.   124 |  1,2,4         1024 |  0,1,2,4         10024 |  0,1,2,4
.   125 |  1,2,5         1025 |  0,1,2,5         10025 |  0,1,2,5  .

Reinhard Zumkeller, Rows n = 0..10000 of triangle, flattened
Index entries for sequences related to palindromes

Cf. A262190 (row lengths), A054054 (left edge), A047813 (right edge), A136522, A002113.

Cf. A031298, A218978.

import Data.List (inits, tails, nub, sort)
a262188 n k = a262188_tabf !! n !! k
a262188_row n = a262188_tabf !! n
a262188_tabf = map (sort . nub . map (foldr (\d v -> 10 * v + d) 0) .
   filter (\xs -> length xs == 1 || last xs > 0 && reverse xs == xs) .
          concatMap (tail . inits) . tails) a031298_tabf

A262188_row(n,b=10)=Set(concat(vector(logint(n+!n,b)+1,m,m=n\=b^(m>1);select(is_A002113,vector(logint(m+!m,b)+1,k,m%b^k))))) \\ M. F. Hasler, Jun 19 2018

A341184 Number of ways to write n as an ordered sum of 5 nonzero decimal palindromes.

Original entry on oeis.org

1, 5, 15, 35, 70, 126, 210, 330, 495, 710, 981, 1310, 1695, 2130, 2605, 3106, 3615, 4110, 4575, 4980, 5311, 5555, 5705, 5760, 5725, 5611, 5435, 5210, 4975, 4735, 4511, 4320, 4170, 4065, 4005, 3986, 4005, 4040, 4100, 4175, 4241, 4310, 4380, 4450, 4520, 4590, 4665, 4740, 4810

Offset: 5

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A002113, A136522, A319470, A341158, A341191, A341192, A341193, A341203, A341204, A341205, A341206, A341207.

nmax = 53; CoefficientList[Series[Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

cp's OEIS Frontend

A341191 Number of ways to write n as an ordered sum of 2 nonzero decimal palindromes.

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A341192 Number of ways to write n as an ordered sum of 3 nonzero decimal palindromes.

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A341193 Number of ways to write n as an ordered sum of 4 nonzero decimal palindromes.

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A047813 Largest palindromic substring of n.

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A062293 Smallest multiple k*n of n which has even digits and is a palindrome or becomes a palindrome when 0's are added on the left (e.g., 10 becomes 010, which is a palindrome).

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A015976 One iteration of Reverse and Add is needed to reach a palindrome.

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A078715 Palindromic Roman numerals.

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A030147 Palindromes in which parity of digits alternates.

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