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

1,3

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.

Links

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

Formula

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

Example

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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A171968 A093474 A030101 * A348363 A379015 A081432

Adjacent sequences: A162739 A162740 A162741 * A162743 A162744 A162745

Keyword

base,easy,nonn,mult

Author

R. J. Mathar, Jul 12 2009

Extensions

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

Status

approved