cp's OEIS Frontend

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

%I A162742 #20 Oct 12 2024 14:35:43
%S A162742 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,
%T A162742 23,15,31,1,39,17,35,9,41,25,33,5,37,21,53,13,45,29,61,3,49,25,51,11,
%U A162742 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
%N A162742 Reverse digits in the binary representation of each prime base in the prime factorization of n.
%C A162742 Base-2 variant of A071786: apply the bit-reversion A030101 to each of the primes in the bases of the prime factorization of n.
%H A162742 Alois P. Heinz, <a href="/A162742/b162742.txt">Table of n, a(n) for n = 1..20000</a>
%F A162742 A161955(n) = A030101(a(n)).
%e A162742 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).
%p A162742 A030101 := proc(n) local dgs ; dgs := convert(n,base,2) ; add( op(-i,dgs)*2^(i-1),i=1..nops(dgs)) ; end:
%p A162742 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:
%t A162742 f[p_, e_] := IntegerReverse[p, 2]^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 24 2023 *)
%o A162742 (Python)
%o A162742 from math import prod
%o A162742 from sympy import factorint
%o A162742 def A162742(n): return prod(int(bin(f)[2:][::-1], 2)**e for f, e in factorint(n).items())
%o A162742 print([A162742(n) for n in range(1, 81)]) # _Michael S. Branicky_, Oct 07 2024
%Y A162742 Cf. A030101, A071786, A161955, A376857 (fixed points).
%K A162742 base,easy,nonn,mult
%O A162742 1,3
%A A162742 _R. J. Mathar_, Jul 12 2009
%E A162742 Cleaned up the definition and corrected the second example - _R. J. Mathar_, Aug 03 2009