A375784 Array read by rows: T(n,k) is the first number with n prime factors (counted with multiplicity) and n occurrences of decimal digit k.

101, 13, 2, 3, 41, 5, 61, 7, 83, 19, 1003, 115, 22, 33, 445, 55, 166, 77, 818, 299, 10002, 1113, 222, 333, 4244, 555, 2666, 777, 8828, 3999, 100002, 11011, 22122, 33332, 4444, 15555, 6666, 75777, 38888, 9999, 1000004, 1011112, 222220, 333330, 444244, 552555, 666366, 777770, 88888, 999996

Offset: 1

Author

Robert Israel, Aug 28 2024

Keywords

• nonn
• base
• tabf

Examples

			T(5,1) = 1011112 = 2^3 * 211 * 599 has 5 prime factors (counted with multiplicity) and 5 1's, and is the first such number.
Array starts
    101      13      2      3     41      5     61      7    83     19
   1003     115     22     33    445     55    166     77   818    299
  10002    1113    222    333   4244    555   2666    777  8828   3999
 100002   11011  22122  33332   4444  15555   6666  75777 38888   9999
1000004 1011112 222220 333330 444244 552555 666366 777770 88888 999996
		

Links

• Robert Israel, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A001222, A002275, A046053, A375760.

Programs

Maple

F:= proc(v, x) local d, y, z, L, S, SS, Cands, t, i, k;
   for d from v do
     Cands:= NULL;
     if x = 0 then SS:= combinat:-choose([$1..d-1], v)
     else SS:= combinat:-choose([$1..d], v)
     fi;
     for S in SS do
       for y from 9^(d-v+1) to 9^(d-v+1)+9^(d-v)-1 do
         L:= convert(y, base, 9)[1..d-v+1];
         L:= map(proc(s) if s < x then s else s+1 fi end proc, L);
         i:= 1;
         t:= 0:
         for k from 1 to d do
           if member(k, S) then t:= t + x*10^(k-1)
           else t:= t + L[i]*10^(k-1); i:= i+1;
           fi;
         od;
         Cands:= Cands, t
     od od;
     Cands:= sort([Cands]);
     for t in Cands do if numtheory:-bigomega(t)=v then return t fi od;
   od
end proc:
for i from 1 to 10 do
  seq(F(i, x), x=0..9)
od;

Mathematica

T[n_, k_]:=Module[{m=2}, While[PrimeOmega[m]!=n||Count[IntegerDigits[m], k]!=n, m++]; m]; Table[T[n, k], {n, 1, 5}, {k, 0, 9}]//Flatten (* Stefano Spezia, Aug 30 2024 *)

Formula

If n - A046053(n) is odd and >= 1, then T(n,k) <= k * A002275(n) * 10^((n - A046053(n) - 1)/2) for k = 2, 3, 5 and 7.

If n - A046053(n) is odd and >= 3, then T(n,8) <= 8 * A002275(n) * 10^((n - A046053(n) - 3)/2).

If n - A046053(n) is even and >= 0, then T(n,1) <= A002275(n) * 10^((n - A046053(n))/2).

If n - A046053(n) is even and >= 2, then T(n,k) <= k * A002275(n) * 10^((n - A046053(n) - 2)/2) for k = 4, 6 and 9.