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A287874 Concatenate prime factorization as in A080670, but write everything in binary.

%I A287874 #33 Apr 11 2021 03:59:04
%S A287874 1,10,11,1010,101,1011,111,1011,1110,10101,1011,101011,1101,10111,
%T A287874 11101,10100,10001,101110,10011,1010101,11111,101011,10111,101111,
%U A287874 10110,101101,1111,1010111,11101,1011101,11111,10101,111011,1010001,101111,10101110,100101
%N A287874 Concatenate prime factorization as in A080670, but write everything in binary.
%C A287874 As in A080670 the prime factorization of n 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 and all binary digits are concatenated.
%C A287874 See A230625 for the terms written in base 10, and for further information (fixed points, trajectories).
%H A287874 Robert Israel, <a href="/A287874/b287874.txt">Table of n, a(n) for n = 1..10000</a>
%F A287874 a(n) = A007088(A230625(n)). - _R. J. Mathar_, Jun 16 2017
%e A287874 a(1) = 1 by convention.
%e A287874 a(2) = 10 (= 2 written in binary).
%e A287874 a(4) = 1010 = concatenate(10,10), since 4 = 2^2 = 10[2] ^ 10[2].
%e A287874 a(6) = 1011 = concatenate(10,11), since 6 = 2*3 = 10[2] * 11[2].
%e A287874 a(8) = 1011 = concatenate(10,11), since 8 = 2^3 = 10[2] ^ 11[2].
%p A287874 f:= proc(n) local F, L, i;
%p A287874     F:= map(op,subs(1=NULL, sort(ifactors(n)[2], (a,b) -> a[1] < b[1])));
%p A287874     F:= map(convert, F, binary);
%p A287874     L:= map(length,F);
%p A287874     L:= ListTools:-Reverse(ListTools:-PartialSums(ListTools:-Reverse(L)));
%p A287874     add(F[i]*10^L[i+1],i=1..nops(F)-1)+F[-1];
%p A287874 end proc:
%p A287874 f(1):= 1:
%p A287874 map(f, [$1..100]); # _Robert Israel_, Jun 20 2017
%t A287874 fn[1] = 1; fn[n_] := FromDigits[Flatten[IntegerDigits[DeleteCases[Flatten[FactorInteger[n]], 1], 2]]];
%t A287874 Table[fn[n], {n, 37}] (* _Robert Price_, Mar 16 2020 *)
%o A287874 (PARI) A287874(n)=if(n>1,fromdigits(concat(apply(binary,select(t->t>1,concat(Col(factor(n))~)))),10),1) \\ _M. F. Hasler_, Jun 21 2017
%o A287874 (Python)
%o A287874 from sympy import factorint
%o A287874 def a(n):
%o A287874     f=factorint(n)
%o A287874     return 1 if n==1 else int("".join(bin(i)[2:] + bin(f[i])[2:] if f[i]!=1 else bin(i)[2:] for i in f))
%o A287874 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 23 2017
%Y A287874 Cf. A080670, A230625, A230626, A230627, A287875.
%K A287874 nonn,base
%O A287874 1,2
%A A287874 _N. J. A. Sloane_, Jun 15 2017
%E A287874 Edited by _M. F. Hasler_, Jun 21 2017