A080670 -id:A080670 - 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

A067599 Decimal encoding of the prime factorization of n: concatenation of prime factors and exponents.

Original entry on oeis.org

21, 31, 22, 51, 2131, 71, 23, 32, 2151, 111, 2231, 131, 2171, 3151, 24, 171, 2132, 191, 2251, 3171, 21111, 231, 2331, 52, 21131, 33, 2271, 291, 213151, 311, 25, 31111, 21171, 5171, 2232, 371, 21191, 31131, 2351, 411, 213171, 431, 22111, 3251, 21231

Offset: 2

Joseph L. Pe, Jan 31 2002

If n has prime factorization p_1^e_1 * ... * p_r^e_r with p_1 < ... < p_r, then its decimal encoding is p_1 e_1...p_r e_r. For example, 15 = 3^1 * 5^1, so has decimal encoding 3151.
Sequence A068633 is a duplicate, up to a conventional initial term a(1)=11.
a(31) = a(177147) = 311. Is there any solution to a(n) = n? - Franklin T. Adams-Watters, Dec 18 2006
The earliest duplicate is a(223) = 2231 = a(12). There is no fixed point below 3*10^6. - M. F. Hasler, Oct 06 2013

			The prime factorization of 24 = 2^3 * 3^1 has corresponding encoding 2331. So a(24) = 2331.
a(42) = 213171 since 42 = 2^1*3^1*7^1. - _Amarnath Murthy_, Feb 27 2002

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A037276, A080670, A112375.

Cf. A027748, A124010.

import Data.Function (on)
a067599 n = read $ foldl1 (++) $
   zipWith ((++) `on` show) (a027748_row n) (a124010_row n) :: Integer
-- Reinhard Zumkeller, Oct 27 2013

with(ListTools): with(MmaTranslator[Mma]): seq(FromDigits(FlattenOnce(ifactors(n)[2])), n=2..46); # Wolfdieter Lang, Aug 16 2014
# second Maple program:
a:= n-> parse(cat(map(i-> i[], sort(ifactors(n)[2]))[])):
seq(a(n), n=2..60);  # Alois P. Heinz, Mar 16 2018

f[n_] := FromDigits[ Flatten[ IntegerDigits[ FactorInteger[ n]]]]; Table[ f[n], {n, 2, 50} ]

A067599(n)=eval(concat(concat([""],concat(Vec(factor(n)~))~))) \\ - M. F. Hasler, Oct 06 2013

Edited by Robert G. Wilson v, Feb 02 2002

Merged contributions from A068633 to here, and minor edits by M. F. Hasler, Oct 06 2013

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

A288818 Number of ways in which one can insert * and ^ into the decimal digits of n to create a valid (see comments) base-ten factorization statement.

Original entry on oeis.org

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

Offset: 1

Hans Havermann, Jun 17 2017

A base-ten factorization statement is valid when it is the product of base-ten powers of (left to right) strictly increasing base-ten primes. A single prime (with or without an exponent) is acceptable. No prime and no exponent may begin with a zero. No exponent may be equal to one.

Excepting 1, a(n) is the number of occurrences of n in A080670.

			a(12) = 0 because there are no valid solutions.
a(1111) = 1 because 11^11 is the only valid statement.
a(7013) = 2 because 7013 and 701^3 are the only solutions.
a(2353797) = 75 because there are 75 valid solutions.
a(13^532*3853*96179) = 1593300019. There are 1593300019 ways of creating valid factorization statements using this 602-digit integer.

Hans Havermann, Table of n, a(n) for n = 1..10000
Hans Havermann, Asterisks and circumflexes
StackExchange, Generating prime factorizations from an integer's digits by inserting * and ^ into the digit sequence

Cf. A288819 (records), A288820 (records' position), A080670.

See the StackExchange link. (* or *)
ric[d_, lp_] := Block[{p, e, i, j, n = Length@d}, If[n == 0, cnt++, If[d[[1]] > 0, Do[p = FromDigits@ Take[d, i]; If[p > lp && PrimeQ@p, ric[Take[d, i - n], p]; Do[e = Take[d, {i + 1, j}]; If[e[[1]] > 0 && e != {1}, ric[Take[d, j - n], p]], {j, i+1, n}]], {i, n}]]]]; a[n_] := (cnt = 0; ric[ IntegerDigits@ n, 1]; cnt); Array[a, 100] (* Giovanni Resta, Jun 19 2017 *)

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

A080695 Concatenation of the prime power factors (with maximal exponent) of n; a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 4, 5, 23, 7, 8, 9, 25, 11, 43, 13, 27, 35, 16, 17, 29, 19, 45, 37, 211, 23, 83, 25, 213, 27, 47, 29, 235, 31, 32, 311, 217, 57, 49, 37, 219, 313, 85, 41, 237, 43, 411, 95, 223, 47, 163, 49, 225, 317, 413, 53, 227, 511, 87, 319, 229, 59, 435, 61, 231, 97, 64

Offset: 1

Vladeta Jovovic, Mar 03 2003

a(n) = n iff n is 1 or a prime power; otherwise, a(n) > n. - Ivan Neretin, May 31 2016

			a(67500) = a(2^2*3^3*5^4) = a(4*27*625) = 427625.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A037276, A067599, A080670, A230305.

Table[FromDigits@Flatten@IntegerDigits[#[[1]]^#[[2]] & /@ FactorInteger[n]], {n, 64}] (* Ivan Neretin, May 31 2016 *)

Edited by Charles R Greathouse IV, Apr 29 2010

A288820 Indices of records in A288818.

Original entry on oeis.org

1, 2, 23, 223, 233, 237, 2237, 2337, 22337, 23137, 23337, 23373, 23797, 223373, 223797, 231373, 233137, 233797, 2233797, 2313797, 2331373, 2333797, 2337379, 2337397, 2353797, 22353797, 22373797, 23137397, 23173797, 23313797, 23353797, 23373797

Offset: 1

Hans Havermann, Jun 17 2017

If one assumes that larger terms begin with 22 or 23, then a(54)-a(64) are: 223373733797, 223537373797, 231337373797, 231353673797, 231353733797, 231373733797, 233137337397, 233537373797, 233753733797, 235313733797, and 235373733797. - Hans Havermann, Jul 31 2017

Giovanni Resta, Table of n, a(n) for n = 1..53

Cf. A288818, A288819 (records' value), A080670.

ric[d_, lp_] := Block[{p, e, i, j, n = Length@d}, If[n == 0, cnt++, If[d[[1]] > 0, Do[p = FromDigits@ Take[d, i]; If[p > lp && PrimeQ@p, ric[Take[d, i - n], p]; Do[e = Take[d, {i + 1, j}]; If[e[[1]] > 0 && e != {1}, ric[Take[d, j - n], p]], {j, i+1, n}]], {i, n}]]]]; a[n_] := (cnt = 0; ric[ IntegerDigits@ n, 1]; cnt); L = {1}; k = 1; bst = 0; While[Length@L < 18, v = a[++k]; If[v > bst, AppendTo[L, k]; bst = v]]; L (* Giovanni Resta, Jun 19 2017 *)

A288819 Record maxima in A288818.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 12, 13, 17, 18, 21, 22, 23, 25, 30, 34, 41, 45, 54, 58, 60, 62, 66, 75, 80, 82, 83, 88, 104, 110, 124, 131, 147, 163, 166, 173, 194, 220, 228, 233, 257, 290, 313, 334, 350, 359, 365, 412, 468, 477, 503, 520

Offset: 1

Hans Havermann, Jun 17 2017

If one assumes that larger records occur at A288818 indices beginning with 22 or 23, then a(54)-a(64) are: 528, 534, 538, 542, 636, 657, 687, 728, 764, 768, and 847. - Hans Havermann, Jul 31 2017

Cf. A288818, A288820 (records' position), A080670.

ric[d_, lp_] := Block[{p, e, i, j, n = Length@d}, If[n == 0, cnt++, If[d[[1]] > 0, Do[p = FromDigits@ Take[d, i]; If[p > lp && PrimeQ@p, ric[Take[d, i - n], p]; Do[e = Take[d, {i + 1, j}]; If[e[[1]] > 0 && e != {1}, ric[Take[d, j - n], p]], {j, i+1, n}]], {i, n}]]]]; a[n_] := (cnt = 0; ric[ IntegerDigits@ n, 1]; cnt); Union@ FoldList[Max, Array[a, 10^6]] (* Michael De Vlieger, Jun 19 2017, after Giovanni Resta at A288818 *)

a(46)-a(53) from Giovanni Resta, Jun 20 2017

A289660 a(n) = A037276(n) - n.

Original entry on oeis.org

0, 0, 0, 18, 0, 17, 0, 214, 24, 15, 0, 211, 0, 13, 20, 2206, 0, 215, 0, 205, 16, 189, 0, 2199, 30, 187, 306, 199, 0, 205, 0, 22190, 278, 183, 22, 2197, 0, 181, 274, 2185, 0, 195, 0, 2167, 290, 177, 0, 22175, 28, 205, 266, 2161, 0, 2279, 456, 2171, 262, 171, 0, 2175, 0, 169, 274, 222158, 448, 2245, 0

Offset: 1

N. J. A. Sloane, Jul 24 2017

The fact that a(n)>0 if n is composite shows that (unlike A080670), A037276 has no nontrivial fixed points (nor any cycles).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A037274, A037276, A080670.

FromDigits@Flatten@IntegerDigits[Table[#1,{#2}]&@@@FactorInteger@#]-#&/@Range@54 (* Vincenzo Librandi, Jul 25 2017 *)

from sympy import factorint
def A289660(n):
    return 0 if n == 1 else int(''.join(map(lambda x: str(x[0])*x[1],sorted(factorint(n).items())))) - n # Chai Wah Wu, Jul 25 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

A067599 Decimal encoding of the prime factorization of n: concatenation of prime factors and exponents.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A288818 Number of ways in which one can insert * and ^ into the decimal digits of n to create a valid (see comments) base-ten factorization statement.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A080695 Concatenation of the prime power factors (with maximal exponent) of n; a(1) = 1 by convention.

Views

Author

Keywords