A290447 -id:A290447 - cp's OEIS frontend

A037276 Start with 1; for n>1, replace n with the concatenation of its prime factors in increasing order.

1, 2, 3, 22, 5, 23, 7, 222, 33, 25, 11, 223, 13, 27, 35, 2222, 17, 233, 19, 225, 37, 211, 23, 2223, 55, 213, 333, 227, 29, 235, 31, 22222, 311, 217, 57, 2233, 37, 219, 313, 2225, 41, 237, 43, 2211, 335, 223, 47, 22223, 77, 255, 317, 2213, 53, 2333

Offset: 1

N. J. A. Sloane

			If n = 2^3*5^5*11^2 = 3025000, a(n) = 222555551111 (n=2*2*2*5*5*5*5*5*11*11, then remove the multiplication signs).

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 10000 terms from T. D. Noe]
Patrick De Geest, Home Primes
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)
Eric Weisstein's World of Mathematics, Home Prime

Cf. A037274, A048985, A067599, A080670, A084796. Different from A073646.

Cf. also A027746, A289660 (a(n)-n).

a037276 = read . concatMap show . a027746_row
-- Reinhard Zumkeller, Apr 03 2012

# This is for n>1
read("transforms") ;
A037276 := proc(n)
    local L,p ;
    L := [] ;
    for p in ifactors(n)[2] do
        L := [op(L),seq(op(1,p),i=1..op(2,p))] ;
    end do:
    digcatL(L) ;
end proc: # R. J. Mathar, Oct 29 2012

co[n_, k_] := Nest[Flatten[IntegerDigits[{#, n}]] &, n, k - 1]; Table[FromDigits[Flatten[IntegerDigits[co @@@ FactorInteger[n]]]], {n, 54}] (* Jayanta Basu, Jul 04 2013 *)
FromDigits@ Flatten@ IntegerDigits[Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Range@ 54 (* Michael De Vlieger, Jul 14 2015 *)

a(n)={ n<4 & return(n); for(i=1,#n=factor(n)~, n[1,i]=concat(vector(n[2,i],j,Str(n[1,i])))); eval(concat(n[1,]))}  \\ M. F. Hasler, Jun 19 2011

from sympy import factorint
def a(n):
    f=factorint(n)
    l=sorted(f)
    return 1 if n==1 else int("".join(str(i)*f[i] for i in l))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 23 2017

A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 1, 6, 36, 216, 1296

Offset: 0

J. H. Conway, Dec 03 2007

We take 0^0 = 1.
The fixed points are in A135385.
For 1-digit or 2-digit numbers this is the same as A075877. - R. J. Mathar, Mar 28 2012
a(A221221(n)) = A133048(A221221(n)) = A222493(n). - Reinhard Zumkeller, May 27 2013

			20 -> 2^0 = 1,
21 -> 2^1 = 2,
24 -> 2^4 = 16,
39 -> 3^9 = 19683,
623 -> 6^2*3 = 108,
etc.

N. J. A. Sloane, Table of n, a(n) for n = 0..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. A075877, A133501 (number of steps to reach fixed point), A133502, A135385 (the conjectured list of fixed points), A135384 (numbers which converge to 2592). For records see A133504, A133505; for the fixed points that are reached when this map is iterated starting at n, see A287877.
Cf. also A133048 (powerback), A031346 and A003001 (persistence).
Cf. also A031298, A007376.

a133500 = train . reverse . a031298_row where
   train []       = 1
   train [x]      = x
   train (u:v:ws) = u ^ v * (train ws)
-- Reinhard Zumkeller, May 27 2013

powertrain:=proc(n) local a,i,n1,n2,t1,t2; n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=1; for i from 0 to floor(t2/2)-1 do a := a*t1[t2-2*i]^t1[t2-2*i-1]; od: if t2 mod 2 = 1 then a:=a*t1[1]; fi; RETURN(n2*a); end; # N. J. A. Sloane, Dec 03 2007

ptm[n_]:=Module[{idn=IntegerDigits[n]},If[EvenQ[Length[idn]],Times@@( #[[1]]^ #[[2]] &/@Partition[idn,2]),(Times@@(#[[1]]^#[[2]] &/@ Partition[ Most[idn],2]))Last[idn]]]; Array[ptm,70,0] (* Harvey P. Dale, Jul 15 2019 *)

def A133500(n):
    s = str(n)
    l = len(s)
    m = int(s[-1]) if l % 2 else 1
    for i in range(0,l-1,2):
        m *= int(s[i])**int(s[i+1])
    return m # Chai Wah Wu, Jun 16 2017

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.

Original entry on oeis.org

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

A284119 Preperiod (or threshold) of orbit of Post's {00, 1101} tag system applied to the word (100)^n, or -1 if this word has an unbounded trajectory.

Original entry on oeis.org

4, 15, 10, 25, 411, 47, 2128, 853, 372, 2805, 366, 2603, 703, 37912, 612, 127, 998, 2401, 1200, 623, 5280, 1778, 1462, 4346269, 4129, 3241, 7018, 3885, 14632, 7019, 4564, 4277, 147688, 1857, 11120, 81141, 20204, 3847, 116014, 7635, 6488, 5665, 6142, 73515, 5826, 6062, 3781, 7865, 28630

Offset: 1

Jeffrey Shallit, Mar 20 2017

nonn

Post's tag system maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1101 to w and deleting the first three letters.
The empty word is included in the count.
The orbit of the word (100)^n for n=110 dies after 43913328040672 iterations. The longest word in the orbit is 31299218, which appeared at iteration 14392308412264. See also A291792. - Lars Blomberg, Oct 04 2017

			For n = 2 the orbit of (100)^2 = 100100 consists of a preperiod of length 15, followed by a periodic portion of length 6:
Preperiod:
100100,
1001101,
11011101,
111011101,
0111011101,
101110100,
1101001101,
10011011101,
110111011101,
1110111011101,
01110111011101,
1011101110100,
11011101001101,
111010011011101,
0100110111011101,
followed by the period:
(011011101110100,
01110111010000,
1011101000000,
11010000001101,
100000011011101,
0000110111011101)
(repeat)

Lars Blomberg, Table of n, a(n) for n = 1..6075
Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.
Lars Blomberg, Graph of the terms
Lars Blomberg, Graph of preperiod of a(110)
Lars Blomberg, Graph of preperiod of a(4974)
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. A284116, A284121, A292090.

From Lars Blomberg, Apr 20 2018: (Start)
Using Excel, trendlines were created for the preperiod of the Post Tag and Watanabe Tag systems as follows:
A284119: y = 8.6528*x^2.0831, R^2 = 0.478.
A292090: y = 8.5595*x^2.1033, R^2 = 0.472.
Although the error value is rather large, the curves are quite similar. (End)

Edited by N. J. A. Sloane, Jul 29 2017. Added "escape clause" to the definition, Apr 19 2018.

A080670 Literal reading of the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 22, 5, 23, 7, 23, 32, 25, 11, 223, 13, 27, 35, 24, 17, 232, 19, 225, 37, 211, 23, 233, 52, 213, 33, 227, 29, 235, 31, 25, 311, 217, 57, 2232, 37, 219, 313, 235, 41, 237, 43, 2211, 325, 223, 47, 243, 72, 252, 317, 2213, 53, 233, 511, 237, 319, 229, 59, 2235

Offset: 1

Jon Perry, Mar 02 2003

Exponents equal to 1 are omitted and therefore this sequence differs from A067599.
Here the first duplicate (ambiguous) term appears already with a(8)=23=a(6), in A067599 this happens only much later. - M. F. Hasler, Oct 18 2014
The number n = 13532385396179 = 13·53^2·3853·96179 = a(n) is (maybe the first?) nontrivial fixed point of this sequence, making it the first known index of a -1 in A195264. - M. F. Hasler, Jun 06 2017

			8=2^3, which reads 23, hence a(8)=23; 12=2^2*3, which reads 223, hence a(12)=223.

T. D. Noe, Table of n, a(n) for n = 1..10000
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)

Cf. A037276, A067599, A230305, A230625, A027748, A124010, A288818.
See A195330, A195331 for those n for which a(n) is a contraction.
See also home primes, A037271.
See A195264 for what happens when k -> a(k) is repeatedly applied to n.
Partial sums: A287881, A287882.

import Data.Function (on)
a080670 1 = 1
a080670 n = read $ foldl1 (++) $
zipWith (c `on` show) (a027748_row n) (a124010_row n) :: Integer
where c ps es = if es == "1" then ps else ps ++ es
-- Reinhard Zumkeller, Oct 27 2013

ifsSorted := proc(n)
        local fs,L,p ;
        fs := sort(convert(numtheory[factorset](n),list)) ;
        L := [] ;
        for p in fs do
                L := [op(L),[p,padic[ordp](n,p)]] ;
        end do;
        L ;
end proc:
A080670 := proc(n)
        local a,p ;
        if n = 1 then
                return 1;
        end if;
        a := 0 ;
        for p in ifsSorted(n) do
                a := digcat2(a,op(1,p)) ;
                if op(2,p) > 1 then
                        a := digcat2(a,op(2,p)) ;
                end if;
        end do:
        a ;
end proc: # R. J. Mathar, Oct 02 2011
# second Maple program:
a:= 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:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 17 2020

f[n_] := FromDigits[ Flatten@ IntegerDigits[ Flatten[ FactorInteger@ n /. {1 -> {}}]]]; f[1] = 1; Array[ f, 60] (* Robert G. Wilson v, Mar 02 2003 and modified Jul 22 2014 *)

A080670(n)=if(n>1, my(f=factor(n),s=""); for(i=1,#f~,s=Str(s,f[i,1],if(f[i,2]>1, f[i,2],""))); eval(s),1) \\ Charles R Greathouse IV, Oct 27 2013; case n=1 added by M. F. Hasler, Oct 18 2014

A080670(n)=if(n>1,eval(concat(apply(f->Str(f[1],if(f[2]>1,f[2],"")),Vec(factor(n)~)))),1) \\ M. F. Hasler, Oct 18 2014

import sympy
[int(''.join([str(y) for x in sorted(sympy.ntheory.factorint(n).items()) for y in x if y != 1])) for n in range(2,100)] # compute a(n) for n > 1
# Chai Wah Wu, Jul 15 2014

Edited and extended by Robert G. Wilson v, Mar 02 2003

A008892 Aliquot sequence starting at 276.

Original entry on oeis.org

276, 396, 696, 1104, 1872, 3770, 3790, 3050, 2716, 2772, 5964, 10164, 19628, 19684, 22876, 26404, 30044, 33796, 38780, 54628, 54684, 111300, 263676, 465668, 465724, 465780, 1026060, 2325540, 5335260, 11738916, 23117724, 45956820, 121129260, 266485716

Offset: 0

N. J. A. Sloane

nonn

It is an open question whether this sequence ever reaches 0. The trajectory has been calculated to 2145 terms, and is still growing, term 2145 being a 214-digit number (see FactorDB link). - N. J. A. Sloane, Jan 11 2023
The aliquot sequence starting at 306 joins this sequence after one step.
This sequence cannot be extended backwards, since A359132(276) = -1. - N. J. A. Sloane, Jan 10 2023
One can note that the k-tuple abundance of 276 is only 5, since a(6) = 3790 is deficient. On the other hand, the k-tuple abundance of a(8) = 2716 is 164 since a(172) is deficient (see A081705 for definition of k-tuple abundance). - Michel Marcus, Dec 31 2013

K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences. Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
Richard K. Guy, Unsolved Problems in Number Theory, B6.
Richard K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.

Tyler Busby, Table of n, a(n) for n = 0..2157 (terms 0..2127 from Daniel Suteu, terms 2128..2140 from Jeppe Stig Nielsen)
Christophe Clavier, Aliquot Sequences
Christophe Clavier, Trajectory of 276 - the first 1576 terms and their factorizations
Christophe Clavier, Trajectory of 276 - the first 1576 terms and their factorizations [Cached copy]
Wolfgang Creyaufmüller, Lehmer Five
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
FactorDB (factordb.com), Search result for last 20 terms of 276 sequence.
Brady Haran and Ben Sparks, An amazing thing about 276, Numberphile YouTube video, 2024.
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)
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 13.
Paul Zimmermann, Recent information
Index entries for sequences related to aliquot parts.

Cf. A001065, A098007 (length of aliquot sequences).

Cf. A008885 (aliquot sequence starting at 30), ..., A008891 (starting at 180).

f := proc(n) option remember; if n = 0 then 276; else sigma(f(n-1))-f(n-1); fi; end:

NestList[DivisorSigma[1, #] - # &, 276, 50] (* Alonso del Arte, Feb 24 2018 *)

a(n, a=276)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

A039654 a(n) = prime reached by iterating f(x) = sigma(x)-1 starting at n, or -1 if no prime is ever reached.

Original entry on oeis.org

2, 3, 11, 5, 11, 7, 23, 71, 17, 11, 71, 13, 23, 23, 71, 17, 59, 19, 41, 31, 47, 23, 59, 71, 41, 71, 71, 29, 71, 31, 167, 47, 53, 47, 233, 37, 59, 71, 89, 41, 167, 43, 83, 167, 71, 47, 167, 167, 167, 71, 97, 53, 167, 71, 167, 79, 89, 59, 167, 61, 167, 103, 311, 83, 167, 67

Offset: 2

David W. Wilson

nonn

It appears nearly certain that a prime is always reached for n>1.
Since sigma(n) > n for n > 1, and sigma(n) = n + 1 only for n prime, the iteration either reaches a prime and loops there, or grows indefinitely. - Franklin T. Adams-Watters, May 10 2010
Guy (2004) attributes this conjecture to Erdos. See Erdos et al. (1990). - N. J. A. Sloane, Aug 30 2017

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.

Franklin T. Adams-Watters, Table of n, a(n) for n=2..10000
Lucilla Baldini and Josef Eschgfäller, Random functions from coupled dynamical systems, arXiv preprint arXiv:1609.01750 [math.CO], 2016. Mentions the conjecture.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Math Overflow discussion, Does iterating a certain function related to the sums of divisors eventually always result in a prime value?
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. A039655 (the number of steps needed), A039649, A039650, A039651, A039652, A039653, A039656, A291301, A291302, A291776, A291777.
For records see A292112, A292113.
Cf. A177343: number of times the n-th prime occurs in this sequence.
Cf. A292874: least k such that a(k) = prime(n).

f[n_]:=Plus@@Divisors[n]-1;Table[Nest[f,n,6],{n,2,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
f[n_] := DivisorSigma[1,n]-1; Table[FixedPoint[f,n], {n,2,100}] (* T. D. Noe, May 10 2010 *)

a(n)=local(m);if(n<2,0,while((m=sigma(n)-1)!=n,n=m);n) \\ Franklin T. Adams-Watters, May 10 2010

A039654(n)=n>1&&until(n==n=sigma(n)-1,);n \\ M. F. Hasler, Sep 25 2017

Contingency for no prime reached added by Franklin T. Adams-Watters, May 10 2010

Changed escape value from 0 to -1 to be consistent with several related sequences. - N. J. A. Sloane, Aug 31 2017

A135385 Fixed points of the map m -> powertrain(m) (see A133500 for definition).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2592, 24547284284866560000000000

Offset: 1

J. H. Conway and N. J. A. Sloane, Dec 11 2007, Dec 19 2007

Probably there are no other terms. There are no other terms below 10^100.

			Under the powertrain map of A133500, 2^5*3^4 = 2592 -> 2^5*9^2 = 2592.
2^46*3^6*5^10*7^2 = 24547284284866560000000000 -> 2^4*5^4*7^2*8^4*2^8*4^8*6^6*5^6*0^0*0^0*0^0*0^0*0^0 = 24547284284866560000000000.

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)

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

A291784 a(n) = (psi(n) + phi(n))/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 14, 13, 15, 16, 16, 17, 21, 19, 22, 22, 23, 23, 28, 25, 27, 27, 30, 29, 40, 31, 32, 34, 35, 36, 42, 37, 39, 40, 44, 41, 54, 43, 46, 48, 47, 47, 56, 49, 55, 52, 54, 53, 63, 56, 60, 58

Offset: 1

N. J. A. Sloane, Sep 02 2017

This is (A001615 + A000010)/2. It is easy to see that this is always an integer.
If n is a power of a prime (including 1 and primes), then a(n) = n, and in any other case a(n) > n. - M. F. Hasler, Sep 09 2017
If n is in A006881, then a(n)=n+1. - Robert Israel, Feb 10 2019

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41 (page 96 of 2nd ed., pages 147ff of 3rd ed.).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Marcin Mazur and Bogdan V. Petrenko, Generalizations of Arnold's version of Euler's theorem for matrices, Japanese Journal of Mathematics, 5:183-189, 2010.
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. A000010, A001615, A006881, A291785, A291786, A291787, A291788.

f:= proc(n) local P, p;
  P:= numtheory:-factorset(n);
  n*(mul((p-1)/p, p=P) + mul((p+1)/p, p=P))/2
end proc:
map(f, [$1..100]); # Robert Israel, Feb 10 2019

psi[n_] := If[n == 1, 1, n*Times @@ (1 + 1/FactorInteger[n][[All, 1]])];
a[n_] := (psi[n] + EulerPhi[n])/2;
Array[a, 100] (* Jean-François Alcover, Feb 25 2019 *)

A291784(n)=(eulerphi(n)+n*sumdivmult(n,d,issquarefree(d)/d))\2 \\ M. F. Hasler, Sep 03 2017

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 21/(4*Pi^2) = 0.531936... . - Amiram Eldar, Dec 05 2023

cp's OEIS Frontend

A037276 Start with 1; for n>1, replace n with the concatenation of its prime factors in increasing order.

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A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^b*c^d* ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.

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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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A284119 Preperiod (or threshold) of orbit of Post's {00, 1101} tag system applied to the word (100)^n, or -1 if this word has an unbounded trajectory.

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A080670 Literal reading of the prime factorization of n.

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A008892 Aliquot sequence starting at 276.

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A039654 a(n) = prime reached by iterating f(x) = sigma(x)-1 starting at n, or -1 if no prime is ever reached.

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A133500 The powertrain or power train map: Powertrain(n): if abcd... is the decimal expansion of a number n, then the powertrain of n is the number n' = a^bc^d ..., which ends in an exponent or a base according as the number of digits is even or odd. a(0) = 0 by convention.