A368683 a(n) is the first semiprime k such that there are as many semiprimes as possible in the n numbers starting at k.

4, 9, 33, 33, 91, 118, 213, 213, 213, 213, 143095, 143095, 10496839, 237520705, 10630829, 7225144283, 54712989442, 54712989442

Offset

1,1

Links

Table of n, a(n) for n=1..18.

Example

For n = 5, there are at most 4 semiprimes in 5 consecutive numbers (because at least one of the 5 is divisible by 4).  In the 5 consecutive numbers starting at 91 there are 4 semiprimes: 91 = 7 * 13, 93 = 3 * 31, 94 = 2 * 47 and 95 = 5 * 19, and no semiprime < 91 works, so a(5) = 91.

Maple

f:= proc(n)
  local P, r, T, M, v, t, V, W, m, Ts, k, x, L, p, i;
  P:= map(p -> `if`(p^2 <= n, p^2, p), select(isprime, [$1..n/2]));
  r:= ilcm(P);
  T:= combinat:-cartprod([seq([$0..p-1], p=P)]);
  M:= 0;
  while not T[finished] do
    v:= T[nextvalue]();
    t:= chrem(v, P);
    V:= [seq(igcd(t+i, r), i=0..n-1)];
    if V[1] <> 1 and not isprime(V[1]) then next fi;
    W[t]:= select(i -> V[i] = 1 or isprime(V[i]), [$1..n]);
    m:= nops(W[t]);
    if m > M then
       M:= m; Ts:= {t};
    elif m = M then Ts:= Ts union {t}
    fi
  od:
  for k from 0 do
    for v in Ts do
      x:= k*r + v;
      L:= W[v] +~ (x-1);
      if andmap(t -> numtheory:-bigomega(t) = 2, L) then return x fi;
    od;
  od:
end proc:
4, 9, 33, seq(f(n), n=4..17);

Crossrefs

Cf. A001358.

Sequence in context: A119574 A006393 A320920 * A048757 A356825 A173659

Adjacent sequences: A368680 A368681 A368682 * A368684 A368685 A368686

Keyword

nonn,more

Author

Robert Israel, Jan 02 2024

Status

approved