A382996 a(n) is the least number k such that both k and k + s have n prime divisors, counted with multiplicity, where s is the sum of the decimal digits of k.

11, 15, 18, 81, 243, 486, 2976, 25488, 128768, 396864, 911232, 8820864, 69940224, 118462464, 1171768320, 1756943946, 11753349120, 272313556992, 491737042890, 2374758457344, 9766784434176, 22675979501496, 269744252387328, 1546075329527736, 6138628058382336

Offset: 1

Author

Robert Israel, May 06 2025

Keywords

• nonn
• base
• hard

Comments

a(n) <= A383665(n) if A383665(n) exists.

Examples

			a(4) = 81 because 81 has sum of digits 9, both 81 = 3^4 and 81 + 9 = 90 = 2 * 3^2 * 5 have 4 prime divisors, counted with multiplicity, and no number smaller than 81 works.
		

Crossrefs

Cf. A001222, A007953, A062028, A381851, A383665.

Programs

Maple

f:= proc(n) uses priqueue; local pq, t,x,s,p,i;
      initialize(pq);
      insert([-2^n, 2$n], pq);
      do
        t:= extract(pq);
        x:= -t[1];
        s:= convert(convert(x,base,10),`+`);
        if numtheory:-bigomega(x+s) = n then return x fi;
        p:= nextprime(t[-1]);
        for i from n+1 to 2 by -1 while t[i] = t[-1] do
          insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
        od;
      od;
end proc:
map(f, [$1..20]);

PARI

generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(bigomega(m*q+sumdigits(m*q)) == k, listput(list, m*q))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, May 24 2025

Formula

A001222(a(n)) = A001222(A062028(a(n))) = n.

Extensions

a(21)-a(22) from Michael S. Branicky, May 08 2025

a(23)-a(25) from Daniel Suteu, May 24 2025