A230625 Concatenate prime factorization written in binary, convert back to decimal.

1, 2, 3, 10, 5, 11, 7, 11, 14, 21, 11, 43, 13, 23, 29, 20, 17, 46, 19, 85, 31, 43, 23, 47, 22, 45, 15, 87, 29, 93, 31, 21, 59, 81, 47, 174, 37, 83, 61, 93, 41, 95, 43, 171, 117, 87, 47, 83, 30, 86, 113, 173, 53, 47, 91, 95, 115, 93, 59, 349, 61, 95, 119, 22

Offset

1,2

Comments

As in A080670 the prime factorization is written as p1^e1*...*pN^eN (except for exponents eK = 1 which are omitted), with all factors and exponents in binary (cf. A007088). Then "^" and "*" signs are dropped, all binary digits are concatenated, and the result is converted back to decimal (base 10). - M. F. Hasler, Jun 21 2017

The first nontrivial fixed point of this function is 255987. Smaller numbers such that a(a(n)) = n are 1007, 1269; 1503, 3751. See A230627 for further information. - M. F. Hasler, Jun 21 2017

255987 is the only nontrivial fixed point less than 10000000. - Benjamin Knight, May 16 2018

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.

N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

Example

6 = 2*3 = (in binary) 10*11 -> 1011 = 11 in base 10, so a(6) = 11.
20 = 2^2*5 = (in binary) 10^10*101 -> 1010101 = 85 in base 10, so a(20) = 85.

Maple

# take ifsSorted from A080670
A230625 := proc(n)
    local Ldgs, p, eb, pb, b ;
    b := 2;
    if n = 1 then
        return 1;
    end if;
    Ldgs := [] ;
    for p in ifsSorted(n) do
        pb := convert(op(1, p), base, b) ;
        Ldgs := [op(pb), op(Ldgs)] ;
        if op(2, p) > 1 then
            eb := convert(op(2, p), base, b) ;
            Ldgs := [op(eb), op(Ldgs)] ;
        end if;
    end do:
    add( op(e, Ldgs)*b^(e-1), e=1..nops(Ldgs)) ;
end proc:
seq(A230625(n), n=1..30) ; # R. J. Mathar, Aug 05 2017

Mathematica

Table[FromDigits[#, 2] &@ Flatten@ Map[IntegerDigits[#, 2] &, FactorInteger[n] /. {p_, 1} :> {p}], {n, 64}] (* Michael De Vlieger, Jun 23 2017 *)

Prog

(Python)
import sympy
[int(''.join([bin(y)[2:] for x in sorted(sympy.ntheory.factorint(n).items()) for y in x if y != 1]), 2) for n in range(2, 100)] # compute a(n) for n > 1
# Chai Wah Wu, Jul 15 2014
(PARI) a(n) = {if (n==1, return(1)); f = factor(n); s = []; for (i=1, #f~, s = concat(s, binary(f[i, 1])); if (f[i, 2] != 1, s = concat(s, binary(f[i, 2]))); ); subst(Pol(s), x, 2); } \\ Michel Marcus, Jul 15 2014
(PARI) A230625(n)=n>1||return(1); fold((x, y)->if(y>1, x<<logint(y<<1, 2)+y, x), concat(Col(factor(n))~)) \\ M. F. Hasler, Jun 21 2017

Crossrefs

Cf. A080670, A230626, A230627, A287875, A287878.

See A289667 for the base 3 version.

See A291803 for partial sums.

Sequence in context: A213962 A216937 A351629 * A048985 A337183 A348058

Adjacent sequences: A230622 A230623 A230624 * A230626 A230627 A230628

Keyword

nonn,base

Author

N. J. A. Sloane, Oct 27 2013

Extensions

More terms from Chai Wah Wu, Jul 15 2014

Added self-contained definition. - M. F. Hasler, Jun 21 2017

Status

approved