cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A261920 In A261907 but not the sum of two palindromes whose lengths differ by 1.

Original entry on oeis.org

10, 109, 110, 113, 114, 115, 116, 117, 118, 119, 120, 124, 125, 126, 127, 128, 129, 130, 135, 136, 137, 138, 139, 140, 146, 147, 148, 149, 150, 157, 158, 159, 160, 168, 169, 170, 179, 180, 190, 211, 221, 231, 241, 251, 261, 271, 281, 291, 302, 312, 322, 332, 342
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Sep 15 2015

Keywords

Comments

Nonpalindromes which are the sum of two palindromes but are not the sum of two palindromes whose lengths differ by 1.

Examples

			10 = 5+5 is a member, as is 109=101+8.

Crossrefs

Cf. A002113, A261907, A261921.

A035137 Numbers that are not the sum of 2 palindromes (where 0 is considered a palindrome).

Original entry on oeis.org

21, 32, 43, 54, 65, 76, 87, 98, 201, 1031, 1041, 1042, 1051, 1052, 1053, 1061, 1062, 1063, 1064, 1071, 1072, 1073, 1074, 1075, 1081, 1082, 1083, 1084, 1085, 1086, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1099, 1101, 1103, 1104, 1105, 1106, 1107, 1108
Offset: 1

Views

Author

Patrick De Geest, Nov 15 1998

Keywords

Comments

Apparently, every positive number is equal to the sum of at most 3 positive palindromes. - Giovanni Resta, May 12 2013
A260254(a(n)) = 0. - Reinhard Zumkeller, Jul 21 2015
A261675(a(n)) >= 3 (and, conjecturally, = 3). - N. J. A. Sloane, Sep 03 2015
This sequence is infinite. Proof: It is easy to see that 200...01 (with any number of zeros) cannot be the sum of two palindromes. - N. J. A. Sloane, Sep 03 2015
The conjecture that every number is the sum of 3 palindromes fails iff there is a term a(n) such that for all palindromes P < a(n), the difference a(n) - P is also a term of this sequence. - M. F. Hasler, Sep 08 2015
Cilleruelo and Luca (see links) have proved the conjecture that every positive integer is the sum of at most three palindromes (in bases >= 5), and also that the density of those that require three is positive. - Christopher E. Thompson, Apr 14 2016

Links

Crossrefs

Cf. A014091, A014092, A104444.
Cf. A260254, A260255 (complement), A002113, A261906, A261907.
Cf. A319477 (disallowing zero).

Programs

  • Haskell
    a035137 n = a035137_list !! (n-1)
a035137_list = filter ((== 0) . a260254) [0..]
-- Reinhard Zumkeller, Jul 21 2015
  • Maple
    N:= 4: # to get all terms with <= N digits
revdigs:= proc(n) local L,j,nL;
  L:= convert(n,base,10); nL:= nops(L);
  add(L[j]*10^(nL-j),j=1..nL);
end proc;
palis:= $0..9:
for d from 2 to N do
  if d::even then
    palis:= palis, seq(x*10^(d/2)+revdigs(x),x=10^(d/2-1)..10^(d/2)-1)
  else
    palis:= palis, seq(seq(x*10^((d+1)/2)+y*10^((d-1)/2)+revdigs(x),y=0..9),x=10^((d-3)/2)..10^((d-1)/2)-1);
  fi
od:
palis:= [palis]:
A:= Array(0..10^N-1):
A[palis]:= 1:
B:= SignalProcessing:-Convolution(A,A):
select(t -> B[t+1] < 0.5, [$1..10^N-1]); # Robert Israel, Jun 22 2015
  • Mathematica
    palQ[n_]:=FromDigits[Reverse[IntegerDigits[n]]]==n; nn=1108; t={}; Do[i=c=0; While[i<=n && c!=1,If[palQ[i] && palQ[n-i], AppendTo[t,n]; c=1]; i++],{n,nn}]; Complement[Range[nn],t] (* Jayanta Basu, May 12 2013 *)
  • PARI
    is_A035137(n)={my(k=0);!until(n<2*k=nxt(k),is_A002113(n-k)&&return)} \\ Uses function nxt() given in A002113. Not very efficient for large n, better start with k=n-A261423(n). Maybe also better use A261423 rather than nxt(). - M. F. Hasler, Jul 21 2015

A261912 Numbers with palindromic order 5.

Original entry on oeis.org

101073, 101082, 101100, 101155, 101199, 102192, 102299, 103275, 103293, 103366, 103399, 103502, 104332, 104342, 104352, 104362, 104372, 104382, 104392, 104499, 104602, 105432, 105442, 105452, 105462, 105472, 105482, 105492, 105493, 105544, 105577, 105599, 105702
Offset: 1

Views

Author

N. J. A. Sloane, Sep 10 2015

Keywords

Comments

See A261913 for definition.
In the Friedman Problem of the Month page, there is a statement by John Hoffman which, if I have interpreted it correctly, asserts that this sequence has only a finite number of terms. However, Chai Wah Wu has extended the sequence out to 10^8, finding 481384 terms, the last one being a(481384) = 99998180. This sequence does not appear to be finite.
The first terms of this sequence are just beyond A109326(5). It can be expected that at least beyond A109326(6) = 1000101024 there will be examples where N-prevpal(N) and N-prevpal(prevpal(N)) are both of order 5; these numbers could be termed to be of order 6, and so on. - M. F. Hasler, Sep 13 2015

Links

Crossrefs

Cf. A002113, A261907, A261910, A261911, A261913, A262528.

Extensions

More terms from Chai Wah Wu, Sep 11 2015 and Sep 12 2015

A261913 The palindromic order of n (defined in Comments).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2
Offset: 0

Views

Author

N. J. A. Sloane, Sep 10 2015

Keywords

Comments

Order 1: palindromes (A002113);
Order 2: not order 1 but is the sum of two palindromes (A261907);
Order 3: not order 1 or 2, but n - previous_palindrome(n) (i.e., n - A261914(n)) gives a number of order 2 (A261910);
Order 4: not order 1, 2, or 3, but subtracting previous_palindrome(previous_palindrome(n)) gives a number of order 2 (A261911);
Order 5: not orders 1, 2, 3, or 4 (A261912).

Links

Crossrefs

Cf. A002113, A261423, A261907, A261910, A261911, A261912, A261914.
Closely related to A261675. See also A088601.

Formula

a(n) = A088601(n). - R. J. Mathar, Feb 14 2023

A261917 Numbers which in base 5 are the sum of two palindromes but are not palindromes themselves.

Original entry on oeis.org

5, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 53, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86, 87, 89, 90, 91, 92, 94, 95, 96, 97, 99
Offset: 1

Views

Author

N. J. A. Sloane, Sep 13 2015

Keywords

Links

Crossrefs

Cf. A029952 (palindromes), A261907 (base 10 analog), A261918.

A261921 Nonpalindromes which are the sum of two palindromes whose lengths differ by 1.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 96, 97, 100, 101, 102, 103, 104, 105, 106, 107, 108, 112, 122, 123, 132
Offset: 1

Views

Author

David Applegate and N. J. A. Sloane, Sep 15 2015

Keywords

Comments

More than the usual number of terms are displayed in order to show that this is different from A261907.

Examples

			12=11+1 and 100=99+1 are members.
10 is not a member since it is only the sum of two palindromes (5+5) whose lengths are equal.

Crossrefs

Cf. A002113. A subsequence of A261907. A261920 shows the differences.

A262173 Number of nonnegative integers < 10^n that are palindromes or the sum of 2 palindromes.

Original entry on oeis.org

1, 10, 92, 991, 8012, 90970, 733052, 8861377, 68729295, 875790193
Offset: 0

Views

Author

Chai Wah Wu, Sep 18 2015

Keywords

Crossrefs

Cf. A260255, A261906, A261907.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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