A368683 - OEIS

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

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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}

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