A175240 Call any positive integer that is a palindrome when written in binary a "binary palindrome". a(n) = the smallest binary palindrome such that a(n)*(the n-th binary palindrome) is not a binary palindrome.

27, 5, 5, 9, 5, 17, 5, 3, 5, 33, 3, 7, 5, 65, 9, 5, 3, 15, 3, 3, 5, 129, 3, 5, 3, 27, 3, 5, 5, 257, 17, 5, 3, 5, 5, 3, 3, 27, 3, 3, 3, 5, 3, 3, 5, 513, 3, 9, 3, 5, 3, 3, 3, 27, 7, 3, 3, 5, 3, 5, 5, 1025, 33, 5, 3, 9, 5, 3, 3, 5, 5, 5, 3, 3, 3, 3, 3, 27, 3, 3, 5, 3, 3, 3, 3, 5, 3, 3, 3, 5, 3, 3, 5

Offset

3,1

Comments

There are no palindromes that work for a(1) and a(2), since the first positive binary palindromes are 0 and 1.

Links

Robert Israel, Table of n, a(n) for n = 3..10000

Formula

A006995(n)*a(n) = A175241(n), a non-palindrome when written in binary.

Example

For n=7, A006995(7) = 15 (1111 in binary). And checking 15*A006995(i) for i>=0, we get 15*0=0, 15*1=15, 15*3=45 that belong to A006995, but 15*5=75 does not, so a(7)=5.

Maple

bp:= proc(n) local L; L:= convert(n, base, 2); L = ListTools:-Reverse(L) end proc:
Bp:= select(bp, [$0..10^6]): nBp:= nops(Bp):
f:= proc(n) local i;
  for i from 3 to nBp do
    if not bp(Bp[i]*Bp[n]) then return Bp[i] fi
  od;
  FAIL
end proc:
map(f, [$3..100]); # Robert Israel, Jan 09 2023

Prog

(PARI) a(n) = my(p=A006995(n), k=1); while(is_A006995(p*A006995(k)), k++); A006995(k); \\ using A006995 PARI codes; Michel Marcus, Jan 09 2023
(Python)
from itertools import count, islice, product
def is_bin_pal(n): return (b:=bin(n)[2:]) == b[::-1]
def bin_pals(): # generator of positive binary palindromes in base 10
    yield 1
    digits, midrange = 2, [[""], ["0", "1"]]
    for digits in count(2):
        for p in product("01", repeat=digits//2-1):
            left = "1"+"".join(p)
            for middle in midrange[digits%2]:
                yield int(left + middle + left[::-1], 2)
def agen(): # generator of terms
    g = bin_pals(); next(g)
    for n in count(3):
        bn = next(g)
        yield next(k for k in bin_pals() if not is_bin_pal(k*bn))
print(list(islice(agen(), 93))) # Michael S. Branicky, Jan 09 2023

Crossrefs

Cf. A006995, A175241.

Sequence in context: A040712 A040714 A040711 * A204877 A040709 A218014

Adjacent sequences: A175237 A175238 A175239 * A175241 A175242 A175243

Keyword

base,nonn

Author

Leroy Quet, Mar 11 2010

Extensions

Extended by Ray Chandler, Mar 13 2010

Offset 3 from Michel Marcus, Jan 09 2023

Status

approved