A254317 a(n) is the least number k such that the number of distinct digits in the prime factorization of k is n (counting terms of the form p^1 as p).

1, 6, 26, 102, 510, 3210, 22890, 153690, 1507290, 15618090

Offset: 1

Author

Michel Lagneau, Jan 28 2015

Keywords

• nonn
• base
• fini
• full

Comments

Write k as product of primes raised to powers; then a(n) is the least number k such that the total number of distinct digits in the product representation of k (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) is equal to n. The first term a(1)= 1 by convention. The sequence is complete.

Property: all exponents are equal to 1 (see the examples below).

Examples

			a(1)  =        1;
a(2)  =        6 = 2*3            and A254315(6)        =  2;
a(3)  =       26 = 2*13           and A254315(26)       =  3;
a(4)  =      102 = 2*3*17         and A254315(102)      =  4;
a(5)  =      510 = 2*3*5*17       and A254315(510)      =  5;
a(6)  =     3210 = 2*3*5*107      and A254315(3210)     =  6;
a(7)  =    22890 = 2*3*5*7*109    and A254315(22890)    =  7;
a(8)  =   153690 = 2*3*5*47*109   and A254315(153690)   =  8;
a(9)  =  1507290 = 2*3*5*47*1069  and A254315(1507290)  =  9;
a(10) = 15618090 = 2*3*5*487*1069 and A254315(15618090) = 10.
		

Crossrefs

Cf. A043537, A254315.

Programs

Maple

with(ListTools):
for n from 2 to 10 do:
  ii:=0:
  for k from 2 to 10^9 while(ii=0)do:
    n0:=length(k):lst:={}:x0:=ifactors(k):
    y:=Flatten(x0[2]):z:=convert(y,set):
    z1:=z minus {1}:nn0:=nops(z1):
     for m from 1 to nn0 do :
      t1:=convert(z1[m],base,10):z2:=convert(t1,set):
      lst:=lst union z2:
     od:
     nn1:=nops(lst):
     if nn1=n then ii:=1:printf ( "%d %d \n",n,k):
     else
     fi:
  od :
od:

Mathematica

f[n_] := Block[{pf = FactorInteger@ n, i}, Length@ DeleteDuplicates@ Flatten@ IntegerDigits@ Rest@ Flatten@ Reap@ Do[If[Last[pf[[i]]] == 1, Sow@ First@ pf[[i]], Sow@ FromDigits@ Flatten[IntegerDigits /@ pf[[i]]]], {i, Length@ pf}]]; b = -1; Flatten@ Last@ Reap@ Do[a = f[n]; If[a > b, Sow[n]; b = a], {n, 10^6}] (* Michael De Vlieger, Jan 29 2015 *)
With[{s = Array[CountDistinct@ Flatten@ IntegerDigits[FactorInteger[#] /. {p_, e_} /; e == 1 :> {p}] &, 10^6]}, Map[FirstPosition[s, #][[1]] &, Range@ Max@ s]] (* Michael De Vlieger, Nov 03 2017 *)

PARI

a(n)=for(k=1,10^5,s=[];F=factor(k);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(s,,8)==n,return(k)))
print1(1,", ");for(n=2,7,print1(a(n),", ")) \\ Derek Orr, Jan 30 2015

Extensions

a(10) corrected by Giovanni Resta, Nov 03 2017