A230627 -id:A230627 - cp's OEIS frontend

A195264 Iterate x -> A080670(x) (replace x with the concatenation of the primes and exponents in its prime factorization) starting at n until reach 1 or a prime (which is then the value of a(n)); or a(n) = -1 if a prime is never reached.

1, 2, 3, 211, 5, 23, 7, 23, 2213, 2213, 11, 223, 13, 311, 1129, 233, 17, 17137, 19

Offset: 1

N. J. A. Sloane, Sep 14 2011, based on discussions on the Sequence Fans Mailing List by Alonso del Arte, Franklin T. Adams-Watters, D. S. McNeil, Charles R Greathouse IV, Sean A. Irvine, and others

J. H. Conway offered $1000 for a proof or disproof for his conjecture that every number eventually reaches a 1 or a prime - see OEIS50 link. - N. J. A. Sloane, Oct 15 2014
However, James Davis has discovered that a(13532385396179) = -1. This number D = 13532385396179 = (1407*10^5+1)*96179 = 13*53^2*3853*96179 is clearly fixed by the map x -> A080670(x), and so never reaches 1 or a prime. - Hans Havermann, Jun 05 2017
The number n = 3^6 * 2331961591220850480109739369 * 21313644799483579440006455257 is a near-miss for another nonprime fixed point. Unfortunately here the last two factors only look like primes (they have no prime divisors < 10), but in fact both are composite. - Robert Gerbicz, Jun 07 2017
The number D' = 13^532385396179 maps to D and so is a much larger number with a(D') = -1. Repeating this process (by finding a prime prefix of D') should lead to an infinite sequence of counterexamples to Conway's conjecture. - Hans Havermann, Jun 09 2017
The first 47 digits of D' form a prime P = 68971066936841703995076128866117893410448319579, so if Q denotes the remaining digits of 13^532385396179 then D'' = P^Q is another counterexample. - Robert Gerbicz, Jun 10 2017
This sequence is different from A037274. Here 8 = 2^3 -> 23 (a prime), whereas in A037274 8 = 2^3 -> 222 -> ... -> 3331113965338635107 (a prime). - N. J. A. Sloane, Oct 12 2014
The value of a(20) is presently unknown (see A195265).

			4 = 2^2 -> 22 =2*11 -> 211, prime, so a(4) = 211.
9 = 3^2 -> 32 = 2^5 -> 25 = 5^2 -> 52 = 2^2*13 -> 2213, prime, so a(9)=2213.

R. J. Mathar, Table of n, a(n) for n = 1..19
J. H. Conway, Five $1000 Problems, Oct 10 2014 (recorded by Bill Cheswick).
Alonso del Arte and Sean A. Irvine, Table of n, a(n) for n = 1..1000
Hans Havermann, Table of n, a(n) for n = 1..10000 (includes links to lengthy (>40) and unknown-outcome evolutions, and a list of unfactored composites in the unknowns' last step)
Hans Havermann, Table of n, a(n) for n = 1..10000 (includes links to lengthy (>40) and unknown-outcome evolutions, and a list of unfactored composites in the unknowns' last step) [Cached copy, pdf version as of May 08 2018, with permission]
Hans Havermann, 13532385396179 precursors
Hans Havermann, 13532385396179 precursors [Cached copy, pdf version as of May 08 2018, with permission]
OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences, Problem Session 2, Oct 10 2014, J. H. Conway, Five $1000 Problems (starting at about 06.44). This sequence is mentioned in the fifth problem, starting at around 19:30.
Tony Padilla and Brady Haran, 13532385396179, Numberphile video, 2017
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)
Doron Zeilberger, Videos of Talks Delivered in SLOANE75/OEIS50 DIMACS Conference on Challenges of Identifying Integer Sequences (see Problem Session 2, Oct 10 2014)

A variant of the home primes, A037271. Cf. A080670, A195265 (trajectory of 20), A195266 (trajectory of 105), A230305, A084318. A230627 (base-2), A290329 (base-3)

f[1] := 1; f[n_] := Block[{p = Flatten[FactorInteger[n]]}, k = Length[p]; While[k > 0, If[p[[k]] == 1, p = Delete[p, k]]; k--]; FromDigits[Flatten[IntegerDigits[p]]]]; Table[FixedPoint[f, n], {n, 19}] (* Alonso del Arte, based on the program for A080670, Sep 14 2011 *)
fn[n_] := FromDigits[Flatten[IntegerDigits[DeleteCases[Flatten[
FactorInteger[n]], 1]]]];
Table[NestWhile[fn, n, # != 1 && ! PrimeQ[#] &], {n, 19}] (* Robert Price, Mar 15 2020 *)

a(n)={n>1 && while(!ispseudoprime(n), n=A080670(n));n} \\ M. F. Hasler, Oct 12 2014

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

A230626 Iterate the map x -> A230625(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, 3, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 1, 4, 0, 4, 0, 3, 1, 1, 0, 1, 2, 4, 2, 3, 0, 2, 0, 2, 1, 5, 1, 7, 0, 1, 1, 2, 0, 2, 0, 2, 3, 3, 0, 1, 3, 3, 1, 1, 0, 1, 3, 2, 6, 2, 0, 1, 0, 2, 2, 2, 2, 3, 0, 1, 2, 6, 0, 2, 0, 4, 4, 3, 2, 2, 0, 4, 4, 3, 0, 5, 2, 2, 2, 3

Offset: 2

N. J. A. Sloane, Oct 27 2013

David J. Seal found that the number 255987 is fixed by the map described in A230625 (or equally A287874), so a(255987) = -1. - N. J. A. Sloane, Jun 15 2017
I also observe that the numbers 1007 and 1269 are mapped to each other by that map, as are the numbers 1503 and 3751 (see the b-file submitted by Chai Wah Wu for A230625). So a(1007) = a(1269) = a(1503) = a(3751) = -1. - David J. Seal, Jun 16 2017
a(217) = a(255) = a(446) = a(558) = a(717) = a(735) = a(775) = a(945) = a(958) = -1 since the trajectory either converges to (1007,1269) or to (1503,3751). 255987 has several preimages, e.g. a(7^25*31^19) = a(3^28*7^7*19) = a(7^12*31^51) = -1. a(3568) = 74 ending in the prime 318792605899852268194734519209581. - Chai Wah Wu, Jun 16 2017
See A287878 for the trajectory of 234, which ends at a prime at step 103. - N. J. A. Sloane, Jun 18 2017
See A288894 for the trajectory of 3932. - Sean A. Irvine, Jun 18 2017

			Starting at 18: 18 = 2*3^2 = 10*11^10 in binary -> 101110 = 46 = 2*23 = 10*10111 -> 1010111 = 87 = 3*29 = 11*11101 -> 1111101 = 125 = 5^3 = 101^11 -> 10111 = 23, prime, taking 4 steps, so a(18) = 4.

Sean A. Irvine, Table of n, a(n) for n = 2..10000 (terms 2..3931 from Chai Wah Wu)

Analogous to A230305.

Cf. A230625, A230627, A080670, A287875, A287878, A288847, A288894.

fn[n_] := FromDigits[Flatten[IntegerDigits[ReplaceAll[FactorInteger[n], {x_, 1} -> {x}], 2]], 2];
Map[Length, Table[NestWhileList[fn, n, # != 1 && ! PrimeQ[#] &], {n, 2, 40}], {1}] - 1 (* Robert Price, Mar 16 2020 *)

More terms from Chai Wah Wu, Jul 15 2014

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

Original entry on oeis.org

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

N. J. A. Sloane, Jun 15 2017

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).

			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].

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

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

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

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

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

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

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

Edited by M. F. Hasler, Jun 21 2017

A287875 Iterate the map x -> A230625(x) starting at n; sequence gives the first prime (or 1) that is reached, written in base 2, or -1 if no prime is ever reached.

Original entry on oeis.org

1, 10, 11, 11111, 101, 1011, 111, 1011, 10111, 11111, 1011, 101011, 1101, 10111, 11101, 11111011, 10001, 10111, 10011, 11111011, 11111, 101011, 10111, 101111, 101011, 111001111, 11101, 10111, 11101, 1111111, 11111, 11111, 111011, 10111, 101111, 1111110011101, 100101

Offset: 1

N. J. A. Sloane, Jun 15 2017

David J. Seal found that the number 255987 is fixed by the map described in A230625 (or equally A287874), so a(255987) = -1. (In fact 255987 is the smallest composite number that is fixed.) - N. J. A. Sloane, Jun 15 2017
Also observe that the numbers 1007 and 1269 are mapped to each other by that map, as are the numbers 1503 and 3751 (see the b-file submitted by Chai Wah Wu for A230625). So they are smaller composite values with a(n) = -1, though not fixed. - David J. Seal, Jun 16 2017
a(217) = a(255) = a(446) = a(558) = a(717) = a(735) = a(775) = a(945) = a(958) = -1 since the trajectory either converges to (1007,1269) or to (1503,3751). - Chai Wah Wu, Jun 16 2017

Chai Wah Wu, Table of n, a(n) for n = 1..3931

Cf. A230625, A230626, A230627 (where the primes reached are written in base 10).

Table[FromDigits@ IntegerDigits[#, 2] &@ If[n == 1, 1, NestWhile[FromDigits[#, 2] &@ Flatten@ Map[IntegerDigits[#, 2] &, FactorInteger[#] /. {p_, 1} :> {p}] &, n, ! PrimeQ@ # &, {2, 1}]], {n, 37}] (* Michael De Vlieger, Jun 24 2017 *)

Changed the "escape" value from 0 to -1 to be consistent with A195264. - N. J. A. Sloane, Jul 27 2017

A287878 Trajectory of 234 under the map x -> A230625(x).

Original entry on oeis.org

234, 749, 1003, 1147, 2021, 2799, 7479, 7957, 9453, 30601, 36783, 122563, 246885, 978327, 3926821, 5159983, 9838787, 16121775, 62223239, 113838307, 775077957, 1043973167, 2519959611, 8311144305, 31830468841

Offset: 1

N. J. A. Sloane, Jun 17 2017

Thanks to the work of Chai Wah Wu, it is known that all numbers less than 234 have trajectories that end either at a fixed point or in a cycle of length 2. David J. Seal and Sean A. Irvine have also studied these trajectories. See A230625, A230626, A230627 for the latest information.

For numbers less than 12388, all trajectories either end at a fixed point or in a cycle of length 2.

Sean A. Irvine, Table of n, a(n) for n = 1..104

Cf. A230625, A230626, A230627, A287875.

A289667 Concatenate prime factorization written in base 3, convert back to decimal.

Original entry on oeis.org

1, 2, 3, 8, 5, 21, 7, 21, 11, 23, 11, 75, 13, 25, 32, 22, 17, 65, 19, 77, 34, 65, 23, 192, 17, 67, 30, 79, 29, 194, 31, 23, 92, 71, 52, 227, 37, 73, 94, 194, 41, 196, 43, 227, 104, 77, 47, 201, 23, 71, 98, 229, 53, 192, 146, 196, 100, 191, 59, 680, 61, 193, 106, 24

Offset: 1

N. J. A. Sloane, Jul 27 2017

A080670 is the base 10 version, A230625 is the binary version.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A080670, A195264, A230625, A230627.

# take ifsSorted from A080670
A289667 := proc(n)
    local Ldgs, p,eb,pb,b ;
    b := 3;
    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(A289667(n),n=1..30) ; # R. J. Mathar, Aug 05 2017

Table[FromDigits[#, 3] &@ Flatten@ Map[IntegerDigits[#, 3] &, FactorInteger[n] /. {p_, e_} /; p > 0 :> If[e == 1, p, {p, e}]], {n, 64}] (* Michael De Vlieger, Jul 29 2017 *)

a(n) = {if (n==1, return(1)); f = factor(n); s = []; for (i=1, #f~, s = concat(s, digits(f[i, 1], 3)); if (f[i, 2] != 1, s = concat(s, digits(f[i, 2], 3))););  fromdigits(s, 3);} \\ Michel Marcus, Jul 27 2017

More terms from Michel Marcus, Jul 27 2017

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

Original entry on oeis.org

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 *)

A288894 Trajectory of 3932 under the map x -> A230625(x).

Original entry on oeis.org

3932, 11223, 30571, 40521, 125967, 247763, 341915, 779817, 1897841, 6074935, 12439665, 123673573, 158531777, 248736245, 779730449, 4713421981, 6575471649, 4270067827, 5040914531, 1592591247, 2126855121, 3959133009, 16523140695

Offset: 1

Sean A. Irvine, Jun 18 2017

Sean A. Irvine, Table of n, a(n) for n = 1..95

Cf. A230625, A230626, A230627, A287875, A287878.

A288847 Numbers whose trajectories under the map x -> A230625(x) never reach a prime.

Original entry on oeis.org

217, 255, 446, 558, 717, 735, 775, 945, 958, 1007, 1062, 1115, 1269, 1344, 1503, 1984, 2215, 2358, 3003, 3751, 3858, 4131, 4471, 5144, 6174, 6627, 6915, 6923, 7033, 7073, 7139, 7434, 7530, 7778, 8125, 8142, 8239, 8335, 8575, 8967, 9186, 9303, 10040, 10179, 10856, 11907, 12081, 12248

Offset: 1

David J. Seal, Jun 18 2017

Sequence suggested by N. J. A. Sloane on the SeqFan mailing list. These are also the numbers n for which A230626(n) = -1 and A230627(n) = A287875(n) = 0. All currently-listed terms (those <= 3858) enter one of the two loops 1007 <-> 1269 and 1503 <-> 3751.

Further values added (Jun 19 2017) based on Sean A. Irvine's extension of the b-file for A230626.

			217, 255, 945, 1007 and 1269 are in the sequence because under the map x -> A230625(x):
217 = 7*31 = binary 111*11111 -> binary 11111111 = 255
255 = 3*5*17 = binary 11*101*10001 -> binary 1110110001 = 945
945 = 3^3*5*7 = binary 11^11*101*111 -> binary 1111101111 = 1007
1007 = 19*53 = binary 10011*110101 -> binary 10011110101 = 1269
1269 = 3^3*47 = binary 11^11*101111 -> binary 1111101111 = 1007

Cf. A230625, A230626, A230627, A287874, A287875, A287878.

a(43)-a(48) from Chai Wah Wu, Jul 13 2017

cp's OEIS Frontend

A195264 Iterate x -> A080670(x) (replace x with the concatenation of the primes and exponents in its prime factorization) starting at n until reach 1 or a prime (which is then the value of a(n)); or a(n) = -1 if a prime is never reached.

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

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Maple

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A230626 Iterate the map x -> A230625(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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Mathematica

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

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Maple

Mathematica

PARI

Python

Formula

Extensions

A287875 Iterate the map x -> A230625(x) starting at n; sequence gives the first prime (or 1) that is reached, written in base 2, or -1 if no prime is ever reached.

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Extensions

A287878 Trajectory of 234 under the map x -> A230625(x).

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A289667 Concatenate prime factorization written in base 3, convert back to decimal.

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Maple

Mathematica