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

1,1

Comments

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

Links

Table of n, a(n) for n=1..25.

Formula

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

Example

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.

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]);

Prog

(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

Crossrefs

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

Sequence in context: A279422 A333488 A053675 * A031944 A049354 A343823

Adjacent sequences: A382993 A382994 A382995 * A382997 A382998 A382999

Keyword

nonn,base,hard

Author

Robert Israel, May 06 2025

Extensions

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

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

Status

approved