A381851 -id:A381851 - cp's OEIS frontend

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

Robert Israel, May 06 2025

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

			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.

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

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]);

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

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

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

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

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.

Original entry on oeis.org

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

Offset: 2

Zak Seidov and Robert Israel, May 04 2025

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.

			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.

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

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]);

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

A001222(a(n)) = A001222(A062028(a(n))) = A001222(A066568(a(n))) = n.

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

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

cp's OEIS Frontend

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.

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