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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). 2

%I

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

%N 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).

%C 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.

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

%e a(1) = 1;

%e a(2) = 6 = 2*3 and A254315(6) = 2;

%e a(3) = 26 = 2*13 and A254315(26) = 3;

%e a(4) = 102 = 2*3*17 and A254315(102) = 4;

%e a(5) = 510 = 2*3*5*17 and A254315(510) = 5;

%e a(6) = 3210 = 2*3*5*107 and A254315(3210) = 6;

%e a(7) = 22890 = 2*3*5*7*109 and A254315(22890) = 7;

%e a(8) = 153690 = 2*3*5*47*109 and A254315(153690) = 8;

%e a(9) = 1507290 = 2*3*5*47*1069 and A254315(1507290) = 9;

%e a(10) = 15618090 = 2*3*5*487*1069 and A254315(15618090) = 10.

%p with(ListTools):

%p for n from 2 to 10 do:

%p ii:=0:

%p for k from 2 to 10^9 while(ii=0)do:

%p n0:=length(k):lst:={}:x0:=ifactors(k):

%p y:=Flatten(x0[2]):z:=convert(y,set):

%p z1:=z minus {1}:nn0:=nops(z1):

%p for m from 1 to nn0 do :

%p t1:=convert(z1[m],base,10):z2:=convert(t1,set):

%p lst:=lst union z2:

%p od:

%p nn1:=nops(lst):

%p if nn1=n then ii:=1:printf ( "%d %d \n",n,k):

%p else

%p fi:

%p od :

%p od:

%t 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 *)

%t 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 *)

%o (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)))

%o print1(1,", ");for(n=2,7,print1(a(n),", ")) \\ _Derek Orr_, Jan 30 2015

%Y Cf. A043537, A254315.

%K nonn,base,fini,full

%O 1,2

%A _Michel Lagneau_, Jan 28 2015

%E a(10) corrected by _Giovanni Resta_, Nov 03 2017

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Last modified July 11 23:26 EDT 2020. Contains 335652 sequences. (Running on oeis4.)