A265509 -id:A265509 - cp's OEIS frontend

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.

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

A265559 Smallest base-2 palindrome m >= n, 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, 10101, 10101, 10101, 10101, 11011, 11011, 11011, 11011, 11011, 11011, 11111, 11111, 11111, 11111, 100001, 100001, 101101, 101101, 101101, 101101, 101101, 101101, 101101, 101101, 101101, 101101, 101101, 101101, 110011

Offset: 0

N. J. A. Sloane, Dec 10 2015

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

See A206914 for the values of m written in base 10.

ispal:= proc(n) global b; # test if n is base-b pallindrome
  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, write in base 10
big10:=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 return(m); fi;
                       od;
lprint("no solution in big10 for n = ", n);
end proc;
# find min pal >= n, write in base 10
bigb:=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); mb:=add(t2[i]*10^(i-1), i=1..nops(t2)); return(mb); fi;
                       od;
lprint("no solution in big10 for n = ", n);
end proc;
b:=2;
[seq(big10(n),n=0..144)]; # A206914
[seq(bigb(n),n=0..144)]; # A265559

b2pal[n_]:=Module[{m=n},While[IntegerDigits[m,2]!=Reverse[IntegerDigits[m,2]],m++]; FromDigits[ IntegerDigits[m,2]]]; Array[b2pal,50,0] (* Harvey P. Dale, Feb 25 2024 *)

A265512 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 3.

Original entry on oeis.org

0, 1, 2, 0, 11, 11, 0, 11, 22, 0, 101, 101, 0, 111, 111, 0, 121, 121, 0, 101, 202, 0, 111, 212, 0, 121, 222, 0, 1001, 1001, 0, 1001, 1001, 0, 1001, 1001, 0, 1001, 1001, 0, 1111, 1111, 0, 1111, 1111, 0, 1001, 1001, 0, 1111, 1111, 0, 1221, 1221, 0, 1001, 2002, 0, 1001, 2002, 0, 1001, 2002, 0, 1001, 2002, 0, 1111, 2112

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.

A265513 a(n) = largest base-4 palindrome m <= n such that every base-4 digit of m is <= the corresponding digit of n; m is written in base 10.

Original entry on oeis.org

0, 1, 2, 3, 0, 5, 5, 5, 0, 5, 10, 10, 0, 5, 10, 15, 0, 17, 17, 17, 0, 21, 21, 21, 0, 25, 25, 25, 0, 29, 29, 29, 0, 17, 34, 34, 0, 21, 38, 38, 0, 25, 42, 42, 0, 29, 46, 46, 0, 17, 34, 51, 0, 21, 38, 55, 0, 25, 42, 59, 0, 29, 46, 63, 0, 65, 65, 65, 0, 65, 65, 65, 0, 65, 65, 65, 0, 65, 65, 65, 0, 65, 65, 65, 0, 85, 85

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.

A265514 a(n) = largest base-4 palindrome m <= n such that every base-4 digit of m is <= the corresponding digit of n; m is written in base 4.

Original entry on oeis.org

0, 1, 2, 3, 0, 11, 11, 11, 0, 11, 22, 22, 0, 11, 22, 33, 0, 101, 101, 101, 0, 111, 111, 111, 0, 121, 121, 121, 0, 131, 131, 131, 0, 101, 202, 202, 0, 111, 212, 212, 0, 121, 222, 222, 0, 131, 232, 232, 0, 101, 202, 303, 0, 111, 212, 313, 0, 121, 222, 323, 0, 131, 232, 333, 0, 1001, 1001, 1001, 0, 1001, 1001, 1001, 0

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.

A265515 a(n) = largest base-5 palindrome m <= n such that every base-5 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, 0, 6, 6, 6, 6, 0, 6, 12, 12, 12, 0, 6, 12, 18, 18, 0, 6, 12, 18, 24, 0, 26, 26, 26, 26, 0, 31, 31, 31, 31, 0, 36, 36, 36, 36, 0, 41, 41, 41, 41, 0, 46, 46, 46, 46, 0, 26, 52, 52, 52, 0, 31, 57, 57, 57, 0, 36, 62, 62, 62, 0, 41, 67, 67, 67, 0, 46, 72, 72, 72, 0, 26, 52, 78, 78, 0, 31, 57, 83, 83, 0, 36

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.

cp's OEIS Frontend

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.

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Maple

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

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Maple

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

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

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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.

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PARI

A265559 Smallest base-2 palindrome m >= n, written in base 2.

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Maple

Mathematica

A265512 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 3.

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A265513 a(n) = largest base-4 palindrome m <= n such that every base-4 digit of m is <= the corresponding digit of n; m is written in base 10.

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A265514 a(n) = largest base-4 palindrome m <= n such that every base-4 digit of m is <= the corresponding digit of n; m is written in base 4.

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A265515 a(n) = largest base-5 palindrome m <= n such that every base-5 digit of m is <= the corresponding digit of n; m is written in base 10.

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