A162742 Reverse digits in the binary representation of each prime base in the prime factorization of n.

1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 13, 3, 11, 7, 15, 1, 17, 9, 25, 5, 21, 13, 29, 3, 25, 11, 27, 7, 23, 15, 31, 1, 39, 17, 35, 9, 41, 25, 33, 5, 37, 21, 53, 13, 45, 29, 61, 3, 49, 25, 51, 11, 43, 27, 65, 7, 75, 23, 55, 15, 47, 31, 63, 1, 55, 39, 97, 17, 87, 35, 113, 9, 73, 41, 75, 25, 91, 33

Offset: 1

Author

R. J. Mathar, Jul 12 2009

Keywords

• base
• easy
• nonn
• mult

Comments

Base-2 variant of A071786: apply the bit-reversion A030101 to each of the primes in the bases of the prime factorization of n.

Examples

			At n=8=2^3, represent 2 as 10 in binary, reverse 10 to give 1, and recombine as 1^3=1 = a(8). At n=14=2*7 =(10)*(111) in binary, reverse the factors to give (1)*(111)=1*7=7=a(14).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A030101, A071786, A161955, A376857 (fixed points).

Programs

Maple

A030101 := proc(n) local dgs ; dgs := convert(n,base,2) ; add( op(-i,dgs)*2^(i-1),i=1..nops(dgs)) ; end:
A162742 := proc(n) local a,p ; a := 1 ; for p in ifactors(n)[2] do a := a* A030101(op(1,p))^op(2,p) ; od: a; end:

Mathematica

f[p_, e_] := IntegerReverse[p, 2]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

Python

from math import prod
from sympy import factorint
def A162742(n): return prod(int(bin(f)[2:][::-1], 2)**e for f, e in factorint(n).items())
print([A162742(n) for n in range(1, 81)]) # Michael S. Branicky, Oct 07 2024

Formula

A161955(n) = A030101(a(n)).

Extensions

Cleaned up the definition and corrected the second example - R. J. Mathar, Aug 03 2009