A206914 -id:A206914 - cp's OEIS frontend

A265510 a(n) = largest base-2 palindrome m <= 2n+1 such that every base-2 digit of m is <= the corresponding digit of 2n+1; m is written in base 2.

1, 11, 101, 111, 1001, 1001, 1001, 1111, 10001, 10001, 10101, 10101, 10001, 11011, 10101, 11111, 100001, 100001, 100001, 100001, 100001, 100001, 101101, 101101, 100001, 110011, 100001, 110011, 100001, 110011, 101101, 111111, 1000001, 1000001, 1000001, 1000001, 1001001, 1001001, 1001001, 1001001, 1000001, 1000001

Offset: 0

N. J. A. Sloane, Dec 09 2015

a(n) = A007088(A265509(n)). - Reinhard Zumkeller, Dec 11 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..8191

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

Cf. A007088.

a265510 = a007088 . a265509  -- Reinhard Zumkeller, Dec 11 2015

A265525 a(n) = largest base-10 palindrome m <= n such that every base-10 digit of m is <= the corresponding digit of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 09 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

Cf. A002113, A031298, A265558.

a265525 n = a265525_list !! n
a265525_list = f a031298_tabf [[]] where
   f (ds:dss) pss = y : f dss pss' where
     y = foldr (\d v -> 10 * v + d) 0 ys
     (ys:_) = dropWhile (\ps -> not $ and $ zipWith (<=) ps ds) pss'
     pss' = if ds /= reverse ds then pss else ds : pss
-- Reinhard Zumkeller, Dec 11 2015

ispal := proc(n) # test for base-b palindrome
local L, Ln, i;
global b;
L := convert(n, base, b);
Ln := nops(L);
for i to floor(1/2*Ln) do
if L[i] <> L[Ln + 1 - i] then return false end if
end do;
return true
end proc
# find max pal <= n and in base-b shadow of n, write in base 10
under10:=proc(n) global b;
local t1,t2,i,m,sw1,L2;
if n mod b = 0 then return(0); fi;
t1:=convert(n,base,b);
for m from n by -1 to 0 do
if ispal(m) then
t2:=convert(m,base,b);
L2:=nops(t2);
sw1:=1;
for i from 1 to L2 do
if t2[i] > t1[i] then sw1:=-1; break; fi;
od:
if sw1=1 then return(m); fi;
fi;
od;
end proc;
b:=10; [seq(under10(n),n=0..144)]; # Gives A265525

A265543 a(n) = smallest base-2 palindrome m >= n such that every base-2 digit of n is <= the corresponding digit of m; m is written in base 2.

Original entry on oeis.org

0, 1, 11, 11, 101, 101, 111, 111, 1001, 1001, 1111, 1111, 1111, 1111, 1111, 1111, 10001, 10001, 11011, 11011, 10101, 10101, 11111, 11111, 11011, 11011, 11011, 11011, 11111, 11111, 11111, 11111, 100001, 100001, 110011, 110011, 101101, 101101, 111111, 111111, 101101, 101101, 111111, 111111, 101101, 101101, 111111

Offset: 0

N. J. A. Sloane, Dec 09 2015

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

See A206913 for the values of m written in base 10.

ispal:= proc(n) global b; # test if n is base-b palindrome
  local L, Ln, i;
  L:= convert(n, base, b);
  Ln:= nops(L);
for i from 1 to floor(Ln/2) do
if L[i] <> L[Ln+1-i] then return(false); fi;
od:
return(true);
end proc;
# find min pal >= n and with n in base-b shadow, write in base 10
over10:=proc(n) global b;
local t1,t2,i,m,sw1,L1;
t1:=convert(n,base,b);
L1:=nops(t1);
for m from n to 10*n do
if ispal(m) then
   t2:=convert(m,base,b);
   sw1:=1;
   for i from 1 to L1 do
      if t1[i] > t2[i] then sw1:=-1; break; fi;
                      od:
   if sw1=1 then return(m); fi;
fi;
                       od;
lprint("no solution in over10 for n = ", n);
end proc;
# find min pal >= n and with n in base-b shadow, write in base 10
overb:=proc(n) global b;
local t1,t2,i,m,mb,sw1,L1;
t1:=convert(n,base,b);
L1:=nops(t1);
for m from n to 10*n do
if ispal(m) then
   t2:=convert(m,base,b);
   sw1:=1;
   for i from 1 to L1 do
      if t1[i] > t2[i] then sw1:=-1; break; fi;
                      od:
   if sw1=1 then mb:=add(t2[i]*10^(i-1), i=1..nops(t2)); return(mb); fi;
fi;
                       od;
lprint("no solution in over10 for n = ", n);
end proc;
b:=2;
[seq(over10(n),n=0..144)]; # A175298
[seq(overb(n),n=0..144)]; # A265543

sb2p[n_]:=Module[{m=n},While[!PalindromeQ[IntegerDigits[m,2]]|| Min[ IntegerDigits[ m,2]-IntegerDigits[n,2]]<0,m++];FromDigits[ IntegerDigits[ m,2]]]; Array[sb2p,50,0] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 15 2017 *)

A265558 a(n) = smallest base-10 palindrome m >= n such that every base-10 digit of n is <= the corresponding digit of m.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 22, 33, 44, 55, 66, 77, 88, 99, 22, 22, 22, 33, 44, 55, 66, 77, 88, 99, 33, 33, 33, 33, 44, 55, 66, 77, 88, 99, 44, 44, 44, 44, 44, 55, 66, 77, 88, 99, 55, 55, 55, 55, 55, 55, 66, 77, 88, 99, 66, 66, 66, 66, 66, 66, 66, 77, 88, 99, 77, 77, 77, 77, 77, 77, 77, 77, 88, 99, 88, 88

Offset: 0

N. J. A. Sloane, Dec 10 2015

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

Cf. A265525.

Maple
```
For Maple code see A265543.
```

a(n,base=10) = { my (d=digits(n,base)); for (k=1, #d\2, d[k]=d[#d+1-k]=max(d[k],d[#d+1-k])); fromdigits(d,base) } \\ Rémy Sigrist, Jun 24 2022

A265511 a(n) = largest base-3 palindrome m <= n such that every base-3 digit of m is <= the corresponding digit of n; m is written in base 10.

Original entry on oeis.org

0, 1, 2, 0, 4, 4, 0, 4, 8, 0, 10, 10, 0, 13, 13, 0, 16, 16, 0, 10, 20, 0, 13, 23, 0, 16, 26, 0, 28, 28, 0, 28, 28, 0, 28, 28, 0, 28, 28, 0, 40, 40, 0, 40, 40, 0, 28, 28, 0, 40, 40, 0, 52, 52, 0, 28, 56, 0, 28, 56, 0, 28, 56, 0, 28, 56, 0, 40, 68, 0, 40, 68, 0, 28, 56, 0, 40, 68, 0, 52, 80, 0, 82, 82, 0, 82, 82, 0, 82

Offset: 0

N. J. A. Sloane, Dec 09 2015

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

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

F:= proc(n) local L;
  L:= convert(n,base,3);
  if L[1] = 0 then return 0 fi;
  add(min(L[i],L[-i])*3^(i-1),i=1..nops(L))
end proc:
map(F, [$0..100]); # Robert Israel, Jan 13 2020

A265523 a(n) = largest base-9 palindrome m <= n such that every base-9 digit of m is <= the corresponding digit of n; m is written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 10, 10, 10, 10, 10, 10, 10, 10, 0, 10, 20, 20, 20, 20, 20, 20, 20, 0, 10, 20, 30, 30, 30, 30, 30, 30, 0, 10, 20, 30, 40, 40, 40, 40, 40, 0, 10, 20, 30, 40, 50, 50, 50, 50, 0, 10, 20, 30, 40, 50, 60, 60, 60, 0, 10, 20, 30, 40, 50, 60, 70, 70, 0, 10, 20, 30, 40, 50, 60, 70, 80, 0, 82, 82

Offset: 0

N. J. A. Sloane, Dec 09 2015

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

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

F:= proc(n) local L;
  L:= convert(n,base,9);
if L[1] = 0 then return 0 fi;
  add(min(L[i],L[-i])*9^(i-1),i=1..nops(L))
end proc:
map(F, [$0..100]); # Robert Israel, Jan 13 2020

A265526 Largest base-2 palindrome m <= n, written in base 2.

Original entry on oeis.org

0, 1, 1, 11, 11, 101, 101, 111, 111, 1001, 1001, 1001, 1001, 1001, 1001, 1111, 1111, 10001, 10001, 10001, 10001, 10101, 10101, 10101, 10101, 10101, 10101, 11011, 11011, 11011, 11011, 11111, 11111, 100001, 100001, 100001, 100001, 100001, 100001, 100001, 100001, 100001, 100001, 100001, 100001, 101101, 101101, 101101

Offset: 0

N. J. A. Sloane, Dec 09 2015

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

ispal:= proc(n) global b; # test for base-b palindrome
  local L, Ln, i;
  L:= convert(n, base, b);
  Ln:= nops(L);
for i from 1 to floor(Ln/2) do
if L[i] <> L[Ln+1-i] then return(false); fi;
od:
return(true);
end proc;
# find max pal <= n, write in base 10
less10:=proc(n) global b;
local t1,t2,i,m,sw1,L2;
t1:=convert(n,base,b);
for m from n by -1 to 0 do
if ispal(m) then return(m); fi;
                        od;
end proc;
# find max pal <= n, write in base b
lessb:=proc(n) global b;
local t1,t2,i,m,mb,sw1,L2;
t1:=convert(n,base,b);
for m from n by -1 to 0 do
if ispal(m) then
   t2:=convert(m,base,b);
   L2:=nops(t2);
   mb:=add(t2[i]*10^(i-1), i=1..L2); return(mb); fi;
                        od;
end proc;
b:=2;
[seq(less10(n),n=0..100)]; # A206913
[seq(lessb(n),n=0..100)]; # A265526
[seq(less10(2*n),n=0..100)]; # A265527
[seq(lessb(2*n),n=0..100)]; # A265528
b:=10;
[seq(less10(n),n=0..100)]; # A261423

A265527 Largest base-2 palindrome m <= 2n, written in base 10.

Original entry on oeis.org

0, 1, 3, 5, 7, 9, 9, 9, 15, 17, 17, 21, 21, 21, 27, 27, 31, 33, 33, 33, 33, 33, 33, 45, 45, 45, 51, 51, 51, 51, 51, 51, 63, 65, 65, 65, 65, 73, 73, 73, 73, 73, 73, 85, 85, 85, 85, 93, 93, 93, 99, 99, 99, 99, 107, 107, 107, 107, 107, 107, 119, 119, 119, 119, 127, 129, 129, 129, 129, 129, 129, 129, 129, 129, 129, 129

Offset: 0

N. J. A. Sloane, Dec 09 2015

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

A265528 Largest base-2 palindrome m <= 2n, written in base 2.

Original entry on oeis.org

0, 1, 11, 101, 111, 1001, 1001, 1001, 1111, 10001, 10001, 10101, 10101, 10101, 11011, 11011, 11111, 100001, 100001, 100001, 100001, 100001, 100001, 101101, 101101, 101101, 110011, 110011, 110011, 110011, 110011, 110011, 111111, 1000001, 1000001, 1000001, 1000001, 1001001, 1001001, 1001001, 1001001, 1001001, 1001001

Offset: 0

N. J. A. Sloane, Dec 09 2015

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

A265546 a(n) = smallest base-4 palindrome m >= n such that every base-4 digit of n is <= the corresponding base-4 digit of m; m is written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 5, 5, 10, 15, 10, 10, 10, 15, 15, 15, 15, 15, 17, 17, 34, 51, 21, 21, 38, 55, 25, 25, 42, 59, 29, 29, 46, 63, 34, 34, 34, 51, 38, 38, 38, 55, 42, 42, 42, 59, 46, 46, 46, 63, 51, 51, 51, 51, 55, 55, 55, 55, 59, 59, 59, 59, 63, 63, 63, 63, 65, 65, 130, 195, 85, 85, 150, 215, 105, 105, 170, 235, 125, 125, 190

Offset: 0

N. J. A. Sloane, Dec 10 2015

Jonathan Frech, Table of n, a(n) for n = 0..10000

Sequences related to palindromic floor and ceiling: A175298, A206913, A206914, A261423, A262038, and the large block of consecutive sequences beginning at A265509.

isok(m, dn) = {my(dm = digits(m, 4)); if ((Vecrev(dm) == dm) && (#dm == #dn), for (i=1, #dn, if (dn[i] > dm[i], return (0))); return(1););}
a(n) = {my(dn = digits(n, 4), m = n); while (!isok(m, dn), m++); m;} \\ Michel Marcus, Apr 07 2021

cp's OEIS Frontend

A265510 a(n) = largest base-2 palindrome m <= 2n+1 such that every base-2 digit of m is <= the corresponding digit of 2n+1; m is written in base 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A265525 a(n) = largest base-10 palindrome m <= n such that every base-10 digit of m is <= the corresponding digit of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

A265543 a(n) = smallest base-2 palindrome m >= n such that every base-2 digit of n is <= the corresponding digit of m; m is written in base 2.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A265558 a(n) = smallest base-10 palindrome m >= n such that every base-10 digit of n is <= the corresponding digit of m.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

A265511 a(n) = largest base-3 palindrome m <= n such that every base-3 digit of m is <= the corresponding digit of n; m is written in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A265523 a(n) = largest base-9 palindrome m <= n such that every base-9 digit of m is <= the corresponding digit of n; m is written in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A265526 Largest base-2 palindrome m <= n, written in base 2.

Views

Author

Keywords

Crossrefs

Programs

Maple

A265527 Largest base-2 palindrome m <= 2n, written in base 10.

Views

Author

Keywords

Crossrefs

A265528 Largest base-2 palindrome m <= 2n, written in base 2.

Views

Author

Keywords

Crossrefs

A265546 a(n) = smallest base-4 palindrome m >= n such that every base-4 digit of n is <= the corresponding base-4 digit of m; m is written in base 10.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI