A235027 Reverse the bits of prime divisors of n (with 2 -> 2), and multiply together: a(0)=0, a(1)=1, a(2)=2, a(p) = revbits(p) for odd primes p, a(u*v) = a(u) * a(v) for composites.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 25, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 31, 32, 39, 34, 35, 36, 41, 50, 33, 40, 37, 42, 53, 52, 45, 58, 61, 48, 49, 50, 51, 44, 43, 54, 65, 56, 75, 46, 55, 60, 47, 62, 63, 64, 55, 78, 97

Offset: 0

Author

Antti Karttunen, Jan 02 2014

Keywords

• nonn
• base
• mult

Comments

This is not a permutation of integers: a(25) = 25 = 5*5 = a(19) is the first case which breaks the injectivity. However, the first 24 terms are equal with A057889, which is a GF(2)[X]-analog of this sequence and which in contrary to this, is bijective. This stems from the fact that the set of irreducible GF(2)[X] polynomials (A014580) is closed under bit-reversal (A056539), while primes (A000040) are not.

Sequence A290078 gives the positions n where the ratio a(n)/n obtains new record values.

Note, instead of A056539 we could as well use A057889 to reverse the bits of n, and also A030101 when restricted to odd primes.

Examples

			a(33) = a(3*11) = a(3) * a(11) = 3 * 13 = 39 (because 3, in binary '11', stays same when reversed, while 11 (eleven), in binary '1011', changes to '1101' = 13).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to binary expansion of n.
• Index to divisibility sequences.

Crossrefs

A235028 gives the fixed points. A235030 numbers such that n <> a(a(n)), or equally A001222(a(n)) > A001222(n). A235145 the number of iterations needed to reach a fixed point or cycle of 2, A235146 its records.

Cf. also A010051, A020639, A030101, A056539, A057889, A074832, A098957, A091214, A204219, A290078.

Programs

Mathematica

f[p_, e_] := IntegerReverse[p, 2]^e; f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Sep 03 2023 *)

PARI

revbits(n) = fromdigits(Vecrev(binary(n)), 2);
a(n) = {my(f = factor(n)); for (k=1, #f~, if (f[k,1] != 2, f[k,1] = revbits(f[k,1]););); factorback(f);} \\ Michel Marcus, Aug 05 2017

Formula

Completely multiplicative with a(0)=0, a(1)=1, a(p) = A056539(p) for primes p (which maps 2 to 2, and reverses the binary representation of odd primes), and a(u*v) = a(u) * a(v) for composites.

Equally, after a(0)=0, a(p * q * ... * r) = A056539(p) * A056539(q) * ... * A056539(r), for primes p, q, etc., not necessarily distinct.

a(0)=0, a(1)=1, a(n) = A056539(A020639(n)) * a(n/A020639(n)).