A067599 -id:A067599 - cp's OEIS frontend

A066817 Conjectured values of first prime in the orbit f(m), f(f(m)), ..., where f(n) = A067599(n) and m = n-th composite number; or 0 if none exists.

0, 2131, 23, 3224591, 0, 0, 241127117451117479045190960709721125675426733715695733779133596697360781090711425903130196316185995152974660668512820125356019549490226189398938302252287927928254649608061563193945459975102656949618158919173931, 0, 0, 0, 2251, 0, 0, 0, 3224591, 314313643123658229739531, 97211238048939739899395714118873644859466103898031, 0, 46747167851021731, 3224591, 97211238048939739899395714118873644859466103898031, 3141114911731, 5171

Offset: 1

Joseph L. Pe, Feb 01 2002

The terms with 0 value listed above are conjectural. There are no primes < 10^30.
From Sean A. Irvine, Nov 09 2023: (Start)
None of the unresolved cases with n < 50 terminates in a prime < 10^130.
Because the trajectories under f can coalesce certain values are known to be equal even if that value is currently unknown. For example, a(1) = a(13) and a(9) = a(14).
Because of the inclusion of exponents 1 in the concatenation defined by f, terms in the trajectory typically grow quicker than in A195264 or A037274.
(End)

Cf. A002808, A067599, A067600, A037274, A195264.

(* f returns an array encoding the prime factorization of n *) f[ n_] := Module[ {a, l, i, t = {} }, a = FactorInteger[ n]; l = Length[ a]; For[ i = 1, i <= l, i++, t = Append[ t, a[ [ i]][ [ 1]]]; t = Append[ t, a[ [ i]][ [ 2]]]]; t];
(* g returns the concatenation of the elements of its input array *) g[ x_] := Module[ {r = "", m = Length[ x], l}, For[ l = 1, l <= m, l++, r = StringJoin[ r, ToString[ x[ [ l]]]]]; r];
(* h returns an array of the digits of its input int string *) h[ n_] := IntegerDigits[ ToExpression[ n]]
(* j returns the number formed from the digits in its input array *) j[ x_] := Module[ {r = 0, m = Length[ x], t = x, l}, For[ l = 1, l <= m, l++, r = 10*r + t[ [ 1]]; t = Rest[ t]]; r];
(* k composes the previous functions *) k[ n_] := j[ h[ g[ f[ n]]]]
s[ n_] := Module[ {a=n, r=0}, While[ !PrimeQ[ a] && a<10^30, a=k[ a]]; If[ PrimeQ[ a], r=a]; r]; Table[ s[ i], {i, 2, 50}]

Offset changed to 1 by Jinyuan Wang, Jul 30 2020

a(7) and a(17) resolved and missing a(21) inserted by Sean A. Irvine, Nov 09 2023

A067254 Numbers k such that the decimal encoding of the prime factorization of k (A067599) ends in k.

Original entry on oeis.org

11, 8571, 11371, 190911, 12711811, 14713491, 19090911, 71119711, 12531135391, 15311195711, 112717566411, 158318548011, 518914376931, 7292811659931

Offset: 1

Joseph L. Pe, Feb 20 2002

a(13) > 2*10^11. 518914376931, 7292811659931, 19090909090909090911 and prime repunits (A004022) are also terms. - Donovan Johnson, Dec 04 2012
Are there any terms not ending in 1? Equivalently, are any terms also in A070003? - Charles R Greathouse IV, Dec 05 2012
a(15) > 10^13. If exponents equal to 1 are not represented (as in A080670), the corresponding sequence starts with 113113, 31373137, and 533517177839 = 853 * 3517 * 177839. - Giovanni Resta, Jun 26 2017

			The prime factorization of 190911 is 3^1 * 7^1 * 9091^1 with decimal encoding 317190911, which ends in 190911. Hence 190911 is a term of the sequence.

Cf. A004022, A067599, A070003, A080670.

(*returns true if a ends with b, false o.w.*) f[a_, b_] := Module[{c, d, e, g, h, i, r}, r = False; c = ToString[a]; d = ToString[b]; e = StringLength[c]; g = StringPosition[c, d]; h = Length[g]; If[h > 0, i = g[[h]]; If[i[[2]] == e, r = True]]; r]; (*gives the decimal encoding of the prime factorization of n*) g[n_] := FromDigits[Flatten[IntegerDigits[FactorInteger[n]]]]; Do[If[f[g[n], n], Print[n]], {n, 1, 10^6} ]

{a067254(a,b) = local(n,v,k,j); for(n=max(2,a),b,v=factor(n); if(eval(concat(vector(matsize(v)[1],k, concat(vector(matsize(v)[2],j,Str(v[k,j]))))))%(10^length(Str(n)))==n,print1(n,",")))}
a067254(2,2*10^7) \\ Klaus Brockhaus, Feb 22 2002

a(5)-a(7) from Klaus Brockhaus, Feb 22 2002
a(8)-a(10) from Donovan Johnson, Mar 26 2010
a(11)-a(12) from Donovan Johnson, Dec 04 2012
a(13) from Giovanni Resta, Jun 09 2017
a(14) from Giovanni Resta, Jun 26 2017

A245270 Like A067599 but write everything in binary, then display the answer in base 10.

Original entry on oeis.org

5, 7, 10, 11, 47, 15, 11, 14, 91, 23, 87, 27, 95, 123, 20, 35, 94, 39, 171, 127, 183, 47, 95, 22, 187, 15, 175, 59, 763, 63, 21, 247, 355, 191, 174, 75, 359, 251, 187, 83, 767, 87, 343, 235, 367, 95, 167, 30, 182, 483, 347, 107, 95, 375, 191, 487, 379, 119

Offset: 2

Chai Wah Wu, Jul 15 2014

The only fixed point < 10^8 is 470367 = 3^4 * 5807^1. - Christopher Scussel, Apr 28 2025

			24 = 2^3 * 3^1 has binary encoding 10_11_11_1, that is, 95 in decimal.

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Cf. A067599, A230625.

a(n) = {f = factor(n); s = []; for (i=1, #f~, s = concat(s, binary(f[i, 1])); s = concat(s, binary(f[i, 2]));); subst(Pol(s), x, 2);} \\ Michel Marcus, Jul 16 2014

import sympy
[int(''.join([bin(y)[2:] for x in sorted(sympy.ntheory.factorint(n).items()) for y in x]),2) for n in range(2,200)] # compute a(n) for n > 1
# Chai Wah Wu, Jul 15 2014

A066985 Reflective numbers: k such that the decimal encoding of the prime factorization of k (A067599) is palindromic.

Original entry on oeis.org

4, 11, 13, 17, 19, 27, 72, 199, 242, 529, 800, 841, 1151, 1171, 1181, 1303, 1352, 1373, 1469, 1747, 1777, 1787, 1922, 1949, 1979, 1999, 2321, 3125, 3362, 3421, 3887, 3993, 4069, 4096, 4232, 5389, 5766, 6272, 7442, 7961, 7969, 8921, 10021, 10082, 10469

Offset: 1

Joseph L. Pe, Feb 01 2002

			The decimal encoding of the prime factorization 13^1 * 313^1 of 4069 is 1313131, which is palindromic. So 4069 is a term of the sequence.

Cf. A067599, A067598.

f[n_] := Flatten[IntegerDigits[FactorInteger[n]]]; Select[Range[2, 11000], f[ # ] == Reverse[f[ # ]] &]

Edited by Robert G. Wilson v, Feb 08 2002

Offset corrected by Mohammed Yaseen, Jul 17 2023

A067671 The prime factors of n are also prime factors of the decimal encoding (A067599) of the prime factorization of n.

Original entry on oeis.org

4, 16, 21, 27, 36, 64, 256, 288, 648, 729, 1024, 1444, 1458, 1764, 1936, 2304, 3125, 4096, 4361, 5184, 6272, 7688, 8277, 9408, 11664, 16384, 18432, 19683, 22472, 22987, 26244, 28125, 29403, 31199, 41472, 43264, 59577, 65536, 67712, 73008

Offset: 1

Joseph L. Pe, Feb 04 2002

			21 = 3^1 * 7^1 has prime factors 3,7, which are also prime factors of the corresponding decimal encoding 3171 = 3^1 * 7^1 * 151^1.

Cf. A067599.

(*f gives the decimal encoding of the prime factorization of n*) f[n_] := FromDigits[Flatten[IntegerDigits[FactorInteger[n]]]]; (*g gives the list of prime factors of n*) g[n_] := Module[{a, l, t}, a = FactorInteger[n]; l = Length[a]; Table[a[[i]][[1]], {i, 1, l}]];
(*main routine*) j[n] := Module[{l1 = g[n], l2 = g[f[n]]}, (Intersection[l1, l2] == l1)]; Select[Range[2, 10^5], j[ # ] &]

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

Original entry on oeis.org

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

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

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 *)

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

A112375 Concatenation of base and exponent of prime powers.

Original entry on oeis.org

21, 31, 22, 51, 71, 23, 32, 111, 131, 24, 171, 191, 231, 52, 33, 291, 311, 25, 371, 411, 431, 471, 72, 531, 591, 611, 26, 671, 711, 731, 791, 34, 831, 891, 971, 1011, 1031, 1071, 1091, 1131, 112, 53, 1271, 27, 1311, 1371, 1391, 1491, 1511, 1571, 1631, 1671

Offset: 1

Zak Seidov, Dec 04 2005

If n = p^q, where p is prime and q > 0, then p concatenated with q is in the sequence.

Might be a good "puzzle" sequence - guess the rule given the first ten or so terms.

			n = 3 = 3^1, so (3 concatenated with 1) = 31 is a term.

Robert Price, Table of n, a(n) for n = 1..1280

Cf. A112376, A064438, A067599.

Map[FromDigits, Select[Table[FactorInteger[i], {i, 2, 10000}],
Length[#] == 1 &], 2] (* Robert Price, Mar 15 2020 *)

for(n=1,300,fac=factor(n);if(matsize(fac)[1]==1,print1(eval(concat(Str(fac[1,1]),Str(fac[1,2]))),",")))

a(n) = A067599(A246655(n)) = A067599(A000961(n+1)). - M. F. Hasler, Mar 14 2018

Edited and extended by Klaus Brockhaus, Jan 21 2006

cp's OEIS Frontend

A066817 Conjectured values of first prime in the orbit f(m), f(f(m)), ..., where f(n) = A067599(n) and m = n-th composite number; or 0 if none exists.

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A067254 Numbers k such that the decimal encoding of the prime factorization of k (A067599) ends in k.

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A245270 Like A067599 but write everything in binary, then display the answer in base 10.

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A066985 Reflective numbers: k such that the decimal encoding of the prime factorization of k (A067599) is palindromic.

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A067671 The prime factors of n are also prime factors of the decimal encoding (A067599) of the prime factorization of n.

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

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Haskell

Maple

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A080670 Literal reading of the prime factorization of n.

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Haskell

Maple

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Extensions

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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