A383665 a(n) is the least number k such that k, k - s and k + s all have n prime divisors, counted with multiplicity, where s is the sum of the decimal digits of k.

15, 102, 204, 408, 3078, 14496, 88448, 128768, 6857312, 111411968, 844844000, 6059394048, 13384999936, 948305874880, 6373064359936, 186505184249928

Offset: 2

Author

Zak Seidov and Robert Israel, May 04 2025

Keywords

• nonn
• base
• hard
• more

Comments

k - s is always divisible by 9, so a(1) does not exist, and a(2) = 15 is the only semiprime k such that k, k - s and k + s are all semiprimes.

Examples

			a(4) = 204 because 204 has digit sum 6, 204 - 6 = 198 = 2 * 3^2 * 11, 204 = 2^2 * 3 * 17 and 204 + 6 = 210 = 2 * 3 * 5 * 7 all have 4 prime divisors, counted with multiplicity, and 204 is the least number that works.
		

Crossrefs

Cf. A001222, A007953, A062028, A066568, A381851, A382996.

Programs

Maple

f:= proc(n) uses priqueue; local pq, t,x,s,p,i;
      initialize(pq);
      insert([-2^n, 2$n], pq);
      do
        t:= extract(pq);
        x:= -t[1];
        s:= convert(convert(x,base,10),`+`);
        if numtheory:-bigomega(x-s) = n and numtheory:-bigomega(x+s) = n then return x fi;
        p:= nextprime(t[-1]);
        for i from n+1 to 2 by -1 while t[i] = t[-1] do
          insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
        od;
      od;
end proc:
map(f, [$2..14]);

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(bigomega(m*q+s) == k && 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(A062028(a(n))) = A001222(A066568(a(n))) = n.

Extensions

a(15) from Michael S. Branicky, May 08 2025

a(16)-a(17) from Daniel Suteu, May 24 2025