A161955 -id:A161955 - cp's OEIS frontend

A161957 Fixed points of A161955.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 37, 41, 43, 45, 47, 51, 53, 59, 61, 63, 67, 71, 73, 79, 83, 85, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 127, 131, 137, 139, 149, 151, 153, 157, 163, 167, 173, 179, 181, 187, 189, 191, 193, 197, 199

Offset: 1

Tanya Khovanova, Jun 22 2009

Fixed points of the TITO2 operation (the TITO operation in binary): numbers a(n) such that A161955(a(n)) = a(n).

All numbers in the sequence are odd. All odd primes A065091 belong to the sequence.

			95 is in this sequence because 95 = 5*19. Prime factors in binary are: 101 and 10011.
Reversing them we get 101 and 11001. The product of the last two numbers is 1111101, which is
the reverse of the binary representation of 95 (1011111).

T. Khovanova, Turning Numbers Inside Out [From _Tanya Khovanova_, Jul 07 2009]

Cf. A161955, A161594, A161597.

reverseBinPower[{n_, k_}] := FromDigits[Reverse[IntegerDigits[n, 2]], 2]^k fBin[n_] := FromDigits[ Reverse[IntegerDigits[ Times @@ Map[reverseBinPower, FactorInteger[n]], 2]], 2] Select[Range[300], fBin[ # ] == # &]

Comments condensed by R. J. Mathar, Aug 14 2009

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

Original entry on oeis.org

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

R. J. Mathar, Jul 12 2009

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

			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).

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

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

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:

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

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

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

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

cp's OEIS Frontend

A161957 Fixed points of A161955.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions