A175242 a(n) = the number of divisors of n that are palindromes when written in binary.

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 1, 2, 1, 2, 4, 1, 2, 3, 1, 2, 4, 1, 1, 2, 2, 1, 4, 2, 1, 4, 2, 1, 3, 2, 3, 3, 1, 1, 2, 2, 1, 4, 1, 1, 6, 1, 1, 2, 2, 2, 4, 1, 1, 4, 2, 2, 2, 1, 1, 4, 1, 2, 6, 1, 3, 3, 1, 2, 2, 3, 1, 3, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 1, 4, 4, 1, 2, 1, 1, 6, 2, 1, 4, 1, 2, 2, 1, 2, 5, 2, 1, 4, 1, 1, 6

Offset: 1

Author

Leroy Quet, Mar 11 2010

Keywords

• base
• nonn

Examples

			a(3) = 2 since 3 has 2 divisors, 1 and 3, that are palindromes when written in binary: 1 and 11.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006995, A244162.

Programs

Maple

a:= n-> add(`if`(l=ListTools[Reverse](l), 1, 0), l=
        map(Bits[Split], numtheory[divisors](n))):
seq(a(n), n=1..105);  # Alois P. Heinz, Jul 15 2022

Mathematica

palbQ[n_]:=Module[{idn2=IntegerDigits[n,2]},idn2==Reverse[idn2]]; Table[ Count[ Divisors[ n],?(palbQ[#]&)],{n,110}] (* _Harvey P. Dale, Mar 27 2019 *)
a[n_] := DivisorSum[n, 1 &, PalindromeQ @ IntegerDigits[#, 2] &]; Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

PARI

is(n) = my(d=binary(n)); d==Vecrev(d); \\ A006995
a(n) = sumdiv(n, d, is(d)); \\ Michel Marcus, Jul 15 2022

Python

from sympy import divisors
def c(n): b = bin(n)[2:]; return b == b[::-1]
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jul 15 2022

Formula

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A244162 = 2.378795... . - Amiram Eldar, Jan 01 2024

Extensions

Extended by Ray Chandler, Mar 13 2010