A381851 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.

10, 20, 40, 80, 224, 448, 2176, 24640, 98816, 287744, 3771392, 5637632, 6508544, 323903488, 1126252544, 7698939904, 20511260672, 249460531200, 857557762048, 582799458304, 11797582053376, 24614476447744, 591901367468032, 1314105503776768, 5988418763882496

Offset: 2

Author

Robert Israel, May 06 2025

Keywords

• nonn
• base
• hard

Comments

k - s is always divisible by 9, so a(1) does not exist.

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

Examples

			a(4) = 40 because 40 has sum of digits 4, both 40 = 2^3 * 5 and 40 - 4 = 36 = 2^2 * 3^2 have 4 prime divisors, counted with multiplicity, and no number < 40 works.
		

Crossrefs

Cf. A001222, A066568, A007953, A383665, A382996.

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, [$2..21]);

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, my(s=sumdigits(m*q)); if(m*q > s && bigomega(m*q-s) == 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(A066568(a(n))) = n

Extensions

a(22)-a(23) from Michael S. Branicky, May 07 2025

a(24)-a(26) from Daniel Suteu, May 24 2025