A289667 -id:A289667 - cp's OEIS frontend

A290329 Iterate the map x -> A289667(x) starting at n; sequence gives primes reached, or -1 if no prime is ever reached.

2, 3, 71, 5, 71, 7, 71, 11, 23, 11, 5261, 13, 17, 23, 703455573449, 17, 703455573449, 19, 5261, 71, 703455573449, 23, 727, 17, 67, 21544131687786037881228990839860266551231, 79, 29, 21544131687786037881228990839860266551231, 31, 23, 239, 71, 229, 227, 37, 73

Offset: 2

Chai Wah Wu, Jul 27 2017

Base 3 analog of A195264 and A230627.

Chai Wah Wu, Table of n, a(n) for n = 2..47

Cf. A290328, A230627, A289667.

A290329 := proc(n)
    local nitr ;
    nitr := n ;
    while ( not isprime(nitr) and nitr <> 1) do
        nitr := A289667(nitr) ;
    end do:
    return nitr ;
end proc:
seq(A290329(n),n=1..20) ; # R. J. Mathar, Aug 05 2017

Table[NestWhile[FromDigits[#, 3] &@ Flatten@ Map[IntegerDigits[#, 3] &, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[e == 1, p, {p, e}]] &, n, ! PrimeQ@ # &], {n, 2, 38}] (* Michael De Vlieger, Jul 29 2017 *)

A290328 Iterate the map x -> A289667(x) starting at n; sequence gives number of steps to reach a prime, or -1 if no prime is ever reached.

Original entry on oeis.org

0, 0, 4, 0, 3, 0, 3, 1, 1, 0, 7, 0, 2, 2, 20, 0, 19, 0, 5, 2, 19, 0, 4, 1, 1, 52, 1, 0, 51, 0, 1, 2, 1, 2, 1, 0, 1, 4, 51, 0, 2, 0, 1, 20, 5, 0

Offset: 2

Chai Wah Wu, Jul 27 2017

Base 3 analog of A230626. See A290350 for the trajectory of 48.

If a(48) != -1, then a(48) > 120. - Chai Wah Wu, Jul 29 2017

Cf. A289667, A230626, A195264, A290350, A290329.

Table[-1 + Length@ NestWhileList[FromDigits[#, 3] &@ Flatten@ Map[IntegerDigits[#, 3] &, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[e == 1, p, {p, e}]] &, n, ! PrimeQ@ # &], {n, 2, 47}] (* Michael De Vlieger, Jul 29 2017 *)

A290350 Trajectory of 48 under the map x -> A289667(x).

Original entry on oeis.org

48, 201, 310, 1894, 5321, 12706, 19475, 38759, 16151, 23120, 16496, 49145, 108244, 499453, 3436777, 10278262, 33836945, 240512002, 913716328, 939389234, 3295603306, 27693125191, 43752137122, 225022142762, 2016929822398, 21026295155804, 66261353668847, 675731396348989

Offset: 1

Chai Wah Wu, Jul 28 2017

Chai Wah Wu, Table of n, a(n) for n = 1..121

Cf. A289667, A290328, A290329.

NestList[FromDigits[#, 3] &@ Flatten@ Map[IntegerDigits[#, 3] &, FactorInteger[#] /. {p_, e_} /; p > 0 :> If[e == 1, p, {p, e}]] &, 48, 27] (* Michael De Vlieger, Jul 29 2017 *)

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

Original entry on oeis.org

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

N. J. A. Sloane, Oct 27 2013

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

			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.

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)

Cf. A080670, A230626, A230627, A287875, A287878.
See A289667 for the base 3 version.
See A291803 for partial sums.

# 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

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

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

A230625(n)=n>1||return(1);fold((x,y)->if(y>1,x<M. F. Hasler, Jun 21 2017

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

More terms from Chai Wah Wu, Jul 15 2014

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

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A290329 Iterate the map x -> A289667(x) starting at n; sequence gives primes reached, or -1 if no prime is ever reached.

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A290328 Iterate the map x -> A289667(x) starting at n; sequence gives number of steps to reach a prime, or -1 if no prime is ever reached.

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A290350 Trajectory of 48 under the map x -> A289667(x).

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

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