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

1, 10, 11, 1010, 101, 1011, 111, 1011, 1110, 10101, 1011, 101011, 1101, 10111, 11101, 10100, 10001, 101110, 10011, 1010101, 11111, 101011, 10111, 101111, 10110, 101101, 1111, 1010111, 11101, 1011101, 11111, 10101, 111011, 1010001, 101111, 10101110, 100101

Offset

1,2

Comments

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.

See A230625 for the terms written in base 10, and for further information (fixed points, trajectories).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A007088(A230625(n)). - R. J. Mathar, Jun 16 2017

Example

a(1) = 1 by convention.
a(2) = 10 (= 2 written in binary).
a(4) = 1010 = concatenate(10,10), since 4 = 2^2 = 10[2] ^ 10[2].
a(6) = 1011 = concatenate(10,11), since 6 = 2*3 = 10[2] * 11[2].
a(8) = 1011 = concatenate(10,11), since 8 = 2^3 = 10[2] ^ 11[2].

Maple

f:= proc(n) local F, L, i;
    F:= map(op, subs(1=NULL, sort(ifactors(n)[2], (a, b) -> a[1] < b[1])));
    F:= map(convert, F, binary);
    L:= map(length, F);
    L:= ListTools:-Reverse(ListTools:-PartialSums(ListTools:-Reverse(L)));
    add(F[i]*10^L[i+1], i=1..nops(F)-1)+F[-1];
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Jun 20 2017

Mathematica

fn[1] = 1; fn[n_] := FromDigits[Flatten[IntegerDigits[DeleteCases[Flatten[FactorInteger[n]], 1], 2]]];
Table[fn[n], {n, 37}] (* Robert Price, Mar 16 2020 *)

Prog

(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
(Python)
from sympy import factorint
def a(n):
    f=factorint(n)
    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))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 23 2017

Crossrefs

Cf. A080670, A230625, A230626, A230627, A287875.

Sequence in context: A318089 A109280 A181756 * A064841 A286200 A286411

Adjacent sequences: A287871 A287872 A287873 * A287875 A287876 A287877

Keyword

nonn,base

Author

N. J. A. Sloane, Jun 15 2017

Extensions

Edited by M. F. Hasler, Jun 21 2017

Status

approved