A373887 a(n) is the length of the longest arithmetic progression of semiprimes ending in the n-th semiprime.

1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 2, 4, 3, 3, 4, 3, 5, 3, 3, 4, 3, 3, 4, 3, 4, 5, 4, 4, 3, 3, 5, 3, 4, 4, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 5, 3, 3, 4, 5, 4, 4, 3, 4, 3, 4, 4, 4, 4, 3, 4, 5, 4, 4, 3, 4, 4, 4, 5, 3, 5, 6, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 3, 3, 4, 4, 5, 5, 4, 4, 4, 4, 4, 5, 4, 5

Offset: 1

Author

Robert Israel, Aug 10 2024

Keywords

• nonn

Comments

a(n) is the greatest k such that there exists d > 0 such that A001358(n) - j*d is in A001358 for j = 0 .. k-1.

The first appearance of m in this sequence is at n where A001358(n) = A096003(m).

Conjectures: a(n) >= 3 for n >= 16.

Limit_{n -> oo} a(n) = oo.

If A001358(n) is divisible by A000040(m), then a(n) >= A373888(m). In particular, the conjectures above are implied by the corresponding conjectures for A373888. - Robert Israel, Aug 19 2024

Examples

			a(5) = 3 because the 5th semiprime is A001358(5) = 14 and there is an arithmetic progression of 3 semiprimes ending in 14, namely 4, 9, 14, and no such arithmetic progression of 4 semiprimes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A096583, A373888.

Programs

Maple

S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):
f:= proc(n) local s,i,m,d,j;
  m:= 1;
  s:= S[n];
  for i from n-1 to 1 by -1 do
    d:= s - S[i];
    if s - m*d < 4 then return m fi;
    for j from 2 while ListTools:-BinarySearch(S,s-j*d) <> 0 do od;
    m:= max(m, j);
  od;
m;
end proc:
map(f, [$1..100]);