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A364363 a(n) is the greatest number with n prime factors, counted with multiplicity, and no decimal digit occurring more than twice, or -1 if there is no such number.

%I A364363 #20 Aug 02 2023 13:48:20
%S A364363 1,9988776655443322001,9988776655443321002,99887766554433120201,
%T A364363 99887766554433210201,99887766554433221001,99887766554433220101,
%U A364363 99887766554433221010,99887766554433122010,99887766554433220110,99887766554433211020,99887766554433212100,99887766554433221100,99887766554433200112
%N A364363 a(n) is the greatest number with n prime factors, counted with multiplicity, and no decimal digit occurring more than twice, or -1 if there is no such number.
%C A364363 a(n) = -1 for n sufficiently large, as there are only finitely many numbers with no digits occurring more than twice.
%C A364363 a(n) > 0 for n <= 29.
%C A364363 From _Jon E. Schoenfield_, Jul 25 2023: (Start)
%C A364363 a(n) = -1 for all n > 65, since no term has more than 20 digits; all 20-digit terms have digit sum 90 and thus are divisible by 9 and can have no more than log_3(9) + floor(log_2(99887766554433221100/9)) = 2 + 63 = 65 prime factors, counted with multiplicity (as 2^64 * 3^2 is a 21-digit number); and no term with fewer than 20 digits can have more than floor(log_2(9988776655443322100)) = 63 prime factors, counted with multiplicity.
%C A364363 a(n) > 0 for n = 0..54, 56, 57, and 62; for all other n, a(n) = -1. (End)
%H A364363 Jon E. Schoenfield, <a href="/A364363/b364363.txt">Table of n, a(n) for n = 0..65</a>
%e A364363 a(3) = 99887766554433120201 = 3^2 * 11098640728270346689 has 3 prime factors with multiplicity and each digit 0 to 9 occurs twice, and is the largest such number.
%p A364363 nextp:= proc(P) local k,m, newP,PL,PG,iv,i;
%p A364363    m:= nops(P);
%p A364363    for k from m-1 by -1 do
%p A364363      if P[k] > P[k+1] then
%p A364363        PL,PG:= selectremove(`<`,P[k+1..m],P[k]);
%p A364363        iv:= max[index](PL);
%p A364363        return [op(P[1..k-1]),PL[iv],op(sort([op(subsop(iv=P[k], PL)),op(PG)],`>`))]
%p A364363       fi
%p A364363    od
%p A364363 end proc:
%p A364363 V:= Array(0..21): V[0]:= 1:
%p A364363 P:= [seq(i$2,i=9..0,-1)]: count:= 1:
%p A364363 while count < 3 do
%p A364363   x:= add(P[i]*10^(19-i),i=1..19):
%p A364363   w:= numtheory:-bigomega(x);
%p A364363   if w <= 2 and V[w] = 0 then V[w]:= x; count:= count+1; fi;
%p A364363   P:= nextp(P);
%p A364363 od:
%p A364363 P:= [seq(i$2,i=9..0,-1)]:
%p A364363 while count < 22 do
%p A364363   x:= add(P[i]*10^(20-i),i=1..20):
%p A364363   w:= numtheory:-bigomega(x);
%p A364363   if w <= 21 and V[w] = 0 then V[w]:= x; count:= count+1; fi;
%p A364363   P:= nextp(P);
%p A364363 od:
%p A364363 convert(V,list);
%Y A364363 Cf. A001222, A363963.
%K A364363 nonn,base
%O A364363 0,2
%A A364363 _Zak Seidov_ and _Robert Israel_, Jul 20 2023