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

%I A230625 #64 May 21 2018 12:00:04
%S A230625 1,2,3,10,5,11,7,11,14,21,11,43,13,23,29,20,17,46,19,85,31,43,23,47,
%T A230625 22,45,15,87,29,93,31,21,59,81,47,174,37,83,61,93,41,95,43,171,117,87,
%U A230625 47,83,30,86,113,173,53,47,91,95,115,93,59,349,61,95,119,22
%N A230625 Concatenate prime factorization written in binary, convert back to decimal.
%C A230625 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
%C A230625 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
%C A230625 255987 is the only nontrivial fixed point less than 10000000. - _Benjamin Knight_, May 16 2018
%H A230625 Chai Wah Wu, <a href="/A230625/b230625.txt">Table of n, a(n) for n = 1..10000</a>
%H A230625 N. J. A. Sloane, <a href="/A195264/a195264.pdf">Confessions of a Sequence Addict (AofA2017)</a>, slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
%H A230625 N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)
%e A230625 6 = 2*3 = (in binary) 10*11 -> 1011 = 11 in base 10, so a(6) = 11.
%e A230625 20 = 2^2*5 = (in binary) 10^10*101 -> 1010101 = 85 in base 10, so a(20) = 85.
%p A230625 # take ifsSorted from A080670
%p A230625 A230625 := proc(n)
%p A230625     local Ldgs, p,eb,pb,b ;
%p A230625     b := 2;
%p A230625     if n = 1 then
%p A230625         return 1;
%p A230625     end if;
%p A230625     Ldgs := [] ;
%p A230625     for p in ifsSorted(n) do
%p A230625         pb := convert(op(1,p),base,b) ;
%p A230625         Ldgs := [op(pb),op(Ldgs)] ;
%p A230625         if op(2, p) > 1 then
%p A230625             eb := convert(op(2,p),base,b) ;
%p A230625             Ldgs := [op(eb),op(Ldgs)] ;
%p A230625         end if;
%p A230625     end do:
%p A230625     add( op(e,Ldgs)*b^(e-1),e=1..nops(Ldgs)) ;
%p A230625 end proc:
%p A230625 seq(A230625(n),n=1..30) ; # _R. J. Mathar_, Aug 05 2017
%t A230625 Table[FromDigits[#, 2] &@ Flatten@ Map[IntegerDigits[#, 2] &, FactorInteger[n] /. {p_, 1} :> {p}], {n, 64}] (* _Michael De Vlieger_, Jun 23 2017 *)
%o A230625 (Python)
%o A230625 import sympy
%o A230625 [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
%o A230625 # _Chai Wah Wu_, Jul 15 2014
%o A230625 (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
%o A230625 (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
%Y A230625 Cf. A080670, A230626, A230627, A287875, A287878.
%Y A230625 See A289667 for the base 3 version.
%Y A230625 See A291803 for partial sums.
%K A230625 nonn,base
%O A230625 1,2
%A A230625 _N. J. A. Sloane_, Oct 27 2013
%E A230625 More terms from _Chai Wah Wu_, Jul 15 2014
%E A230625 Added self-contained definition. - _M. F. Hasler_, Jun 21 2017