A080670 -id:A080670 - 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

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

A230305 Iterate A080670 starting at n; a(n) = number of steps to reach a prime, or -1 if no prime is ever reached.

Original entry on oeis.org

0, 0, 2, 0, 1, 0, 1, 4, 3, 0, 1, 0, 3, 4, 2, 0, 3, 0

Offset: 2

N. J. A. Sloane, Oct 27 2013

If n is a prime, a(n) = 0.

a(20) is presently unknown - see A195265 for the trajectory.

			9 -> 32 -> 25 -> 52 -> 2213, which is prime, taking 4 steps, so a(9) = 4.

Cf. A080670, A195264 (the prime that is reached), A195265, A067599, A037271 (home primes).

fn[n_] := FromDigits[Flatten[IntegerDigits[DeleteCases[Flatten[FactorInteger[n]], 1]]]];
Map[Length, Table[NestWhileList[fn, n, # != 1 && ! PrimeQ[#] &], {n, 2,
19}], {1}] - 1 (* Robert Price, Mar 16 2020 *)

A195330 Numbers n such that A080670(n) < n.

Original entry on oeis.org

32, 64, 81, 121, 125, 128, 169, 243, 256, 289, 320, 343, 361, 375, 384, 405, 448, 486, 512, 529, 567, 625, 640, 686, 729, 768, 841, 875, 896, 961, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875, 2048, 2187, 2197, 2209, 2401, 2430, 2500, 2560, 2592, 2656, 2662, 2738, 2744, 2752, 2809, 2816, 2848, 2916, 2944, 3008, 3072, 3104, 3125, 3136, 3200, 3328, 3362, 3375, 3392, 3402, 3430, 3456, 3481, 3483, 3584, 3645, 3698, 3712

Offset: 1

N. J. A. Sloane, Sep 15 2011

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 220 terms from Robert Price)

Cf. A080670, A195331, A194263.

b:= proc(n) option remember; `if`(n=1, 1, (l->
      parse(cat(seq(`if`(l[i, 2]=1, l[i, 1], [l[i, 1],
      l[i, 2]][]), i=1..nops(l)))))(sort(ifactors(n)[2])))
    end:
a:= proc(n) option remember; local k; for k from
      `if`(n=1, 0, a(n-1))+1 while b(k)>=k do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 17 2020

A080670 = Cases[Import["https://oeis.org/A080670/b080670.txt", "Table"], {, }][[All, 2]];
Select[Range[3712], A080670[[#]] < # &] (* Robert Price, Mar 17 2020 *)

is(n,f=factor(n))=eval(concat(vector(#f~,i,if(f[i,2]==1, Str(f[i,1]), Str(f[i,1],f[i,2])))))Charles R Greathouse IV, Mar 18 2020

A287881 Partial sums of A080670.

Original entry on oeis.org

1, 3, 6, 28, 33, 56, 63, 86, 118, 143, 154, 377, 390, 417, 452, 476, 493, 725, 744, 969, 1006, 1217, 1240, 1473, 1525, 1738, 1771, 1998, 2027, 2262, 2293, 2318, 2629, 2846, 2903, 5135, 5172, 5391, 5704, 5939, 5980, 6217, 6260, 8471, 8796, 9019, 9066, 9309, 9381, 9633, 9950, 12163, 12216, 12449, 12960

Offset: 1

N. J. A. Sloane, Jun 19 2017

It would be nice to have an estimate of how fast this sequence grows.

Robert Israel observed (see link) that for the first 100000 terms, a(n) is roughly c*n^2*(log n)^2, where c is between 0.2 and 0.25. However, the ratio a(n)/(n log n)^2 does not seem to be converging.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Robert Israel, Graph of a(n)/(n log n)^2 for n <= 100000

Cf. A080670.

G:= proc(n) local L;
    local F;
    F:= sort(ifactors(n)[2],(a,b) -> a[1]Robert Israel, Jun 19 2017

A080670 = Cases[Import["https://oeis.org/A080670/b080670.txt", "Table"], {, }][[All, 2]];
Accumulate[A080670] (* Robert Price, Mar 16 2020 *)

A195265 Trajectory of 20 under iteration of the map x -> A080670(x).

Original entry on oeis.org

20, 225, 3252, 223271, 297699, 399233, 715623, 3263907, 32347303, 160720129, 1153139393, 72171972859, 736728093411, 3245576031137, 11295052366467, 310807934835791, 1789205424940407, 31745337977379983, 1122916740775279751, 7251536377635958081, 151243563319717018007

Offset: 1

N. J. A. Sloane, Sep 14 2011, based on a posting to the Sequence Fans Mailing List by Alonso del Arte

The table that I submitted for A195264 (see the second link here) is still actively maintained by me. That includes all unknown-outcome evolutions starting with numbers up to 10000. If you click in that table on the 'unknown' beside the number 20 it will give you the current state of the evolution of the number 20. There was a bottleneck at Alonso20(102) [= A195265(103); we're using offset zero for our evolutions] involving a 62-digit factor, cracked by "Mathew" in MersenneForum on August 13. Sean A. Irvine subsequently extended that to Alonso20(109). The unfactored composite in Alonso20(110) is 178 digits long. I maintain links to sorted lists of unfactored composites at the bottom of the table. If anyone can factor any of these composites, submit the factorization to factordb.com and I will (eventually) find it; a personal heads-up would of course be appreciated. - Hans Havermann, Oct 27 2013

			20 = 2^2*5 -> 225 = 3^2*5^2 -> 3252 = 2^2*3*271 -> 223271 ...

Hans Havermann, Table of n, a(n) for n = 1..110
Hans Havermann, Table of n, A195264(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)
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. A037274, A080670, A195264, A195266, A230305.

Maple
```
# See A195266
```

A080670[n_] := ToExpression@StringJoin[ToString/@Flatten[DeleteCases[FactorInteger[n], 1, -1]]]; NestWhileList[A080670, i = 1; 20, (PrintTemporary[{i++, #}]; ! PrimeQ[#]) &, 1, 40] (* Wouter Meeussen, Oct 27 2013 *)

Alonso del Arte computed 40 terms, D. S. McNeil extended it to 66 terms, Sean A. Irvine to 70 terms, Hans Havermann (Oct 27 2013) to 110 terms.

A195266 Trajectory of 105 under iteration of the map x -> A080670(x).

Original entry on oeis.org

105, 357, 3717, 32759, 174147, 358049, 379677, 3196661, 13245897, 373120347, 31961239449, 364811643843, 3474632755849, 7148938489519, 19530970872089, 731453184134581, 1069684240583849, 11451757737372871, 18163269379764491, 99042547183388553, 344278174560973471, 71388716692555572127, 148872380947952962303, 1453102458624189919451, 11245912921175349453489

Offset: 1

N. J. A. Sloane, Sep 14 2011, based on a posting to the Sequence Fans Mailing List by Alonso del Arte

Sean A. Irvine, Table of n, a(n) for n = 1..108 (1..40 from _Alonso del Arte_, 41..65 from _D. S. McNeil_, 66..108 from _Sean A. Irvine_)

Cf. A080670, A195264, A194265.

read("transforms");
# insert A080670 here
A195266 := proc(n)
        option remember;
        if n = 1 then
                105;
        else
                A080670(procname(n-1)) ;
        end if;
end proc: # R. J. Mathar, Oct 02 2011

A195331 A080670 applied to A195330(n).

Original entry on oeis.org

25, 26, 34, 112, 53, 27, 132, 35, 28, 172, 265, 73, 192, 353, 273, 345, 267, 235, 29, 232, 347, 54, 275, 273, 36, 283, 292, 537, 277, 312, 210, 373, 355, 254, 285, 113, 372, 236, 293, 412, 357, 573, 287, 432, 354, 211, 37, 133, 472, 74, 2355, 2254, 295, 2534, 2583, 2113, 2372, 2373, 2643, 532, 2811, 2589, 2236, 2723, 2647, 2103, 2597, 55, 2672, 2752, 2813, 2412, 3353, 2653, 2357, 2573, 2733, 592, 3443, 297, 365, 2432, 2729, 612, 2354, 2659, 3447, 2835, 2435, 2661, 2731

Offset: 1

N. J. A. Sloane, Sep 15 2011

A195330 gives the numbers k such that A080670(k) < k; this sequence gives the corresponding values of A080670(k).

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 220 terms from Robert Price)

Cf. A080670, A195330.

b:= proc(n) option remember; `if`(n=1, 1, (l->
      parse(cat(seq(`if`(l[i, 2]=1, l[i, 1], [l[i, 1],
      l[i, 2]][]), i=1..nops(l)))))(sort(ifactors(n)[2])))
    end:
g:= proc(n) option remember; local k; for k from
      `if`(n=1, 0, g(n-1))+1 while b(k)>=k do od; k
    end:
a:= n-> b(g(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 17 2020

A080670 = Cases[Import["https://oeis.org/A080670/b080670.txt", "Table"], {, }][[All, 2]];
A195330 = Cases[Import["https://oeis.org/A195330/b195330.txt", "Table"], {, }][[All, 2]];
Table[A080670[[n]], {n, A195330}](* Robert Price, Mar 17 2020 *)

A287882 a(n) = Sum_{ i <= n, i squarefree} A080670(i).

Original entry on oeis.org

1, 3, 6, 6, 11, 34, 41, 41, 41, 66, 77, 77, 90, 117, 152, 152, 169, 169, 188, 188, 225, 436, 459, 459, 459, 672, 672, 672, 701, 936, 967, 967, 1278, 1495, 1552, 1552, 1589, 1808, 2121, 2121, 2162, 2399, 2442, 2442, 2442, 2665, 2712, 2712, 2712, 2712, 3029, 3029, 3082, 3082, 3593, 3593, 3912

Offset: 1

N. J. A. Sloane, Jun 19 2017

nonn

A lower bound on A287881.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

Cf. A080670, A287881.

A080670 = Cases[Import["https://oeis.org/A080670/b080670.txt", "Table"], {, }][[All, 2]];
Accumulate[Table[If[SquareFreeQ[n], A080670[[n]], 0], {n, 57}] ] (* Robert Price, Mar 16 2020 *)

A288519 a(n) = least k such that A080670(k) begins with n.

Original entry on oeis.org

1, 2, 3, 41, 5, 61, 7, 83, 97, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 2003, 22, 4, 6, 16, 10, 64, 14, 166, 29, 307, 31, 9, 27, 81, 15, 183, 21, 249, 291, 401, 41, 421, 43, 443, 457, 461, 47, 487, 491, 503, 55, 25, 53, 205, 265, 305, 35, 415, 59, 601, 61

Offset: 1

Rémy Sigrist, Jun 10 2017

a(p) <= p for any prime p.

a(n) <= A018800(n) for any n > 0.

			The following table shows the first values from A080670, as well as the terms that can be derived from it:
k       A080670(k)      Derived terms
--      ----------      -------------
1       1               a(1) = 1
2       2               a(2) = 2
3       3               a(3) = 3
4       22              a(22) = 4
5       5               a(5) = 5
6       23              a(23) = 6
7       7               a(7) = 7
8       23              none
9       32              a(32) = 9
10      25              a(25) = 10
11      11              a(11) = 11
12      223             a(223) = 12
13      13              a(13) = 13
14      27              a(27) = 14
15      35              a(35) = 15
16      24              a(24) = 16
17      17              a(17) = 17
18      232             a(232) = 18
19      19              a(19) = 19
20      225             a(225) = 20
21      37              a(37) = 21
22      211             a(211) = 22, a(21) = 22
23      23              none
24      233             a(233) = 24
25      52              a(52) = 25

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 1000000 terms
Rémy Sigrist, PARI program for A288519

Cf. A018800, A080670.

s = Array[Flatten@ Map[IntegerDigits, DeleteCases[ Flatten@ FactorInteger@ #, 1] /. {} -> {1}] &, 10^4]; FromDigits /@ Table[ Function[k, SelectFirst[s, If[Length@# > 0, #[[1, 1]] == 1, False] &@ SequencePosition[#, k] &]]@ IntegerDigits[n], {n, 61}] (* Michael De Vlieger, Jun 11 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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A287874 Concatenate prime factorization as in A080670, but write everything in binary.

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A230305 Iterate A080670 starting at n; a(n) = number of steps to reach a prime, or -1 if no prime is ever reached.

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A195330 Numbers n such that A080670(n) < n.

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A287881 Partial sums of A080670.

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A195265 Trajectory of 20 under iteration of the map x -> A080670(x).

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A195266 Trajectory of 105 under iteration of the map x -> A080670(x).

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A195331 A080670 applied to A195330(n).

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