A141707 -id:A141707 - cp's OEIS frontend

A044051 a(n) = (s(n)+1)/2, where s=A006995 (base-2 palindromes).

1, 2, 3, 4, 5, 8, 9, 11, 14, 16, 17, 23, 26, 32, 33, 37, 43, 47, 50, 54, 60, 64, 65, 77, 83, 95, 98, 110, 116, 128, 129, 137, 149, 157, 163, 171, 183, 191, 194, 202, 214, 222, 228, 236, 248, 256, 257, 281, 293, 317, 323, 347, 359, 383

Offset: 1

Melia Aldridge, ma38(AT)spruce.evansville.edu

A141707(a(n)) = 1. - Reinhard Zumkeller, Apr 20 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006995, A141707.

a044051 = (`div` 2) . (+ 1) . a006995 . (+ 1)
-- Reinhard Zumkeller, Apr 20 2015

Let n >= 3, m=floor(log_2(n)), p=floor((3*2^(m-1)-1)/n); then a(n) = 2^(2*m-2-p) + 1 + p*(1-(-1)^n)*2^(m-1) + (1/2)*Sum_{k=1..m-1-p} (floor((n - (3-p)*2^(m-1))/2^(m-1-k)) mod 2)*(2^k + 2^(2*m-1-p-k)). - Hieronymus Fischer, Feb 18 2012

A141709 Least positive multiple of n which is palindromic in base 2, allowing for leading zeros (or: ignoring trailing zeros).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 12, 65, 14, 15, 16, 17, 18, 513, 20, 21, 66, 2047, 24, 325, 130, 27, 28, 1421, 30, 31, 32, 33, 34, 455, 36, 2553, 1026, 195, 40, 1025, 42, 129, 132, 45, 4094, 4841, 48, 1421, 650, 51, 260, 3339, 54, 165, 56, 513, 2842, 6077, 60, 427, 62

Offset: 1

M. F. Hasler, Jul 17 2008

Even numbers cannot be palindromic in base 2, unless leading zeros are considered (or, equivalently, resp. more precisely, trailing zeros are discarded). This is done in this version of A141708, which therefore does not need to be restricted to odd n as it has been done for A141707 and A141708.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050782, A141707-A141708, A062279, A203070, A000265, A178225.

a141709 n = until ((== 1) . a178225 . a000265) (+ n) n
-- Reinhard Zumkeller, Nov 06 2012

notpalbinQ[i_]:=Module[{id=IntegerDigits[i,2]},While[Last[id]==0,id=Most[id]];id!= Reverse[id]]; lm[n_]:=Module[{k=1},While[notpalbinQ[k n],k++];k n]; Array[lm,70] (* Harvey P. Dale, Dec 28 2011 *)

A141709(n)=forstep(k=n,10^9,n,vecextract(t=binary(k>>valuation(k,2)),"-1..1")-t || return(k))

A178225(A000265(a(n))) = 1. - Reinhard Zumkeller, Nov 06 2012

A141708 Least positive multiple of 2n-1 which is palindromic in base 2.

Original entry on oeis.org

1, 3, 5, 7, 9, 33, 65, 15, 17, 513, 21, 2047, 325, 27, 1421, 31, 33, 455, 2553, 195, 1025, 129, 45, 4841, 1421, 51, 3339, 165, 513, 6077, 427, 63, 65, 1273, 2553, 10437, 73, 975, 231, 1501, 891, 3735, 85, 3219, 2047, 273, 93, 2565, 5917, 99, 23533, 4841, 1365, 107

Offset: 1

M. F. Hasler, Jul 17 2008

Even numbers cannot be palindromic in base 2 (unless leading zeros are considered), that's why we search for odd numbers 2n-1 their smallest multiple k(2n-1) which is palindromic in base 2. Obviously this must always be odd.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050782, A141707, A062279.

a141708 n = a141707 n * (2 * n - 1) -- Reinhard Zumkeller, Apr 20 2015

pal2[n_]:=Module[{k=1},While[IntegerDigits[k n,2] != Reverse[ IntegerDigits[ k n,2]],k++];k n]; pal2/@Range[1,121,2] (* Harvey P. Dale, Feb 29 2012 *)

A141708(n,L=10^9)={ n=2*n-1; forstep(k=1,L,2, binary(k*n)-vecextract(binary(k*n),"-1..1") || return(k*n))}

def binpal(n): b = bin(n)[2:]; return b == b[::-1]
def a(n):
    m = 2*n - 1
    km = m
    while not binpal(km): km += m
    return km
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Mar 20 2022

a(n) = (2n-1)*A141707(n).

cp's OEIS Frontend

A044051 a(n) = (s(n)+1)/2, where s=A006995 (base-2 palindromes).

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A141709 Least positive multiple of n which is palindromic in base 2, allowing for leading zeros (or: ignoring trailing zeros).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A141708 Least positive multiple of 2n-1 which is palindromic in base 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula