A104444 -id:A104444 - cp's OEIS frontend

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

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

Patrick De Geest, Nov 15 1998

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

David W. Wilson, Table of n, a(n) for n = 1..10000
Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv:1602.06208 [math.NT], 2016.
P. De Geest, World!Of Numbers
Hugo Pfoertner, Plot of first 10^6 terms
Eric Weisstein's World of Mathematics, Palindromic Number

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

a035137 n = a035137_list !! (n-1)
a035137_list = filter ((== 0) . a260254) [0..]
-- Reinhard Zumkeller, Jul 21 2015

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

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 *)

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

A329544 Add the odd terms and subtract the even ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 3, 2, 5, 4, 19, 11, 22, 6, 17, 14, 8, 7, 15, 16, 27, 24, 13, 18, 29, 26, 37, 33, 44, 28, 39, 36, 25, 30, 41, 38, 49, 46, 35, 40, 51, 48, 59, 45, 10, 68, 32, 21, 20, 9, 55, 58, 47, 50, 61, 60, 71, 66, 77, 23, 12, 88, 191, 101, 111, 91, 112, 31, 81, 121, 131, 141, 70, 132, 80, 122, 90, 142, 174, 43, 54, 72, 83, 53, 42

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Nov 16 2019

Negative palindromes are not allowed (0 is OK). After 50000 terms, the smallest unused integers are 919, 1020, 1029, 1031, 1038, 1041, 1047, ... and the largest used is 208831. The largest palindrome produced so far is 1000001. Is the sequence a permutation of the integers > 0?
After one million terms, the smallest unused integers are still the seven mentioned (above) for 50000 terms. - Hans Havermann, Nov 27 2019
This sequence is not a permutation of the nonnegative integers because it cannot contain any term of A104444. The value 919 may only appear after a running total equal to 0 (see A083142, A084843). - Rémy Sigrist, Dec 11 2019. There are only two 0's in the first million terms of A329796, at n=12 and n=1002, so the chance that this happens seems slight. On the other hand, the zeros in the base 3 analog, A330314, are more plentiful (see A330325), so further investigation is needed. - Hans Havermann and N. J. A. Sloane, Dec 12 2019

			The sequence starts with 1 which is positive and a palindrome.
1 + 3 = 4 (palindrome). (2 is not allowed because 1 - 2 < 0.)
1 + 3 - 2 = 2 (palindrome)
1 + 3 - 2 + 5 = 7 (palindrome)
1 + 3 - 2 + 5 - 4 = 3 (palindrome)
1 + 3 - 2 + 5 - 4 + 19 = 22 (palindrome)
1 + 3 - 2 + 5 - 4 + 19 + 11 = 33 (palindrome)
1 + 3 - 2 + 5 - 4 + 19 + 11 - 22 = 11 (palindrome), etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..50000.
Hans Havermann, High-resolution graph of 100000 terms showing even/odd distribution
N. J. A. Sloane, Table of n, a(n), A329796(n), A329796(n)/a(n) for n = 1..50000
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Cf. A329545 (same idea, but where the odd integers are subtracted and the even ones are added).
Cf. A002113 (palindromes), A086862 (first differences), A104444, A329796 (running totals), A329797, A329798 (records), A330311 (when n appears).
See also A083142, A084843, A330313, A330314, A330325.

A329544_vec(N,u=1,U,a,s,d)={vector(N,n, a=u; while(bittest(U,a-u)|| Vecrev(d=digits(s-(-1)^a*a))!=d|| (a>s&&!bittest(a,0)),a++); s-=(-1)^a*a; U+=1<<(a-u); while(bittest(U,0), U>>=1; u++);a)} \\ M. F. Hasler, Nov 16 2019

A084843 Numbers n such that no palindrome m > 0 exists with n + m also a palindrome.

Original entry on oeis.org

919, 1020, 1029, 1031, 1038, 1041, 1047, 1051, 1061, 1065, 1071, 1074, 1081, 1091, 1101, 1130, 1131, 1139, 1141, 1148, 1151, 1157, 1161, 1171, 1175, 1181, 1191, 1201, 1231, 1240, 1241, 1249, 1251, 1258, 1261, 1267, 1271, 1281, 1291, 1301, 1314, 1341

Offset: 1

Patrick De Geest, Jun 08 2003

			There is no palindrome x > 0 such that x + 919 (itself coincidentally also palindromic) is a palindrome.

Cf. A002113, A082272, A082273, A104444.

A330311 Where n appears in A329544, or -1 if n never appears.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 13, 12, 45, 40, 7, 56, 18, 11, 14, 15, 10, 19, 6, 44, 43, 8, 55, 17, 28, 21, 16, 25, 20, 29, 63, 42, 23, 175, 34, 27, 22, 31, 26, 35, 30, 80, 75, 24, 39, 33, 48, 37, 32, 49, 36, 85, 79, 76, 46, 187, 130, 47, 38, 51, 50, 89, 84, 110, 109

Offset: 1

N. J. A. Sloane, Dec 10 2019

nonn

a(919) is unknown. If it is not -1, it is greater than 10^6 (see A329544).

N. J. A. Sloane, Table of n, a(n) for n = 1..918

Cf. A104444, A329544, A329797, A328798.

a(A104444(n)) = -1. - Rémy Sigrist, Dec 11 2019

A329545 After a(1) = 1, add the even terms and subtract the odd ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.

Original entry on oeis.org

1, 2, 3, 4, 18, 11, 5, 16, 13, 7, 6, 14, 15, 26, 22, 33, 17, 28, 25, 36, 35, 9, 8, 58, 55, 44, 46, 10, 20, 30, 73, 77, 66, 24, 40, 50, 103, 79, 68, 34, 23, 81, 48, 47, 80, 83, 72, 54, 43, 85, 52, 49, 38, 37, 70, 64, 53, 87, 32, 27, 60, 57, 90, 12, 45, 59, 92, 42, 75, 61, 94, 62, 95, 63, 74, 69, 194

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Nov 16 2019

Negative palindromes are not allowed. After 50000 terms, the smallest unused integers are 964, 1020, 1029, 1031, 1038, 1041, 1047, 1051, ... and the largest used is 173410. The largest palindrome produced so far is 309903. Is the sequence a permutation of the integers > 0?

a(63411) = 964. Rémy Sigrist's comment in A329544 shows that terms in A104444 are not in the sequence. Conjecture: Sequence is a permutation of positive integers not in A104444. - Chai Wah Wu, Dec 11 2019

			The sequence starts with 1, smallest positive integer.
1 + 2 = 3 (palindrome)
1 + 2 - 3 = 0 (palindrome)
1 + 2 - 3 + 4 = 1 (palindrome)
1 + 2 - 3 + 4 + 18 = 22 (palindrome)
1 + 2 - 3 + 4 + 18 - 11 = 11 (palindrome)
1 + 2 - 3 + 4 + 18 - 11 - 5 = 6 (palindrome)
1 + 2 - 3 + 4 + 18 - 11 - 5 + 16 = 22 (palindrome), etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..49999

Cf. A329544 (same idea, but where the odd integers are added and the even ones are subtracted).
Cf. A002113 (palindromes), A086862 (first differences of palindromes).
Cf. A104444.

A329545_vec(N, u=1, U, a, s=2, d)={vector(N, n, a=u; while(bittest(U, a-u)|| Vecrev(d=digits(s+(-1)^a*a))!=d|| (a>s&&bittest(a, 0)), a++); s+=(-1)^a*a; U+=1<<(a-u); while(bittest(U, 0), U>>=1; u++); a)} \\ M. F. Hasler, Nov 16 2019

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

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A329544 Add the odd terms and subtract the even ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.

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A084843 Numbers n such that no palindrome m > 0 exists with n + m also a palindrome.

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A330311 Where n appears in A329544, or -1 if n never appears.

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A329545 After a(1) = 1, add the even terms and subtract the odd ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.

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