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

%I A373887 #38 Aug 21 2024 05:40:17
%S A373887 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,
%T A373887 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,
%U A373887 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
%N A373887 a(n) is the length of the longest arithmetic progression of semiprimes ending in the n-th semiprime.
%C A373887 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.
%C A373887 The first appearance of m in this sequence is at n where A001358(n) = A096003(m).
%C A373887 Conjectures: a(n) >= 3 for n >= 16.
%C A373887 Limit_{n -> oo} a(n) = oo.
%C A373887 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
%H A373887 Robert Israel, <a href="/A373887/b373887.txt">Table of n, a(n) for n = 1..10000</a>
%e A373887 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.
%p A373887 S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):
%p A373887 f:= proc(n) local s,i,m,d,j;
%p A373887   m:= 1;
%p A373887   s:= S[n];
%p A373887   for i from n-1 to 1 by -1 do
%p A373887     d:= s - S[i];
%p A373887     if s - m*d < 4 then return m fi;
%p A373887     for j from 2 while ListTools:-BinarySearch(S,s-j*d) <> 0 do od;
%p A373887     m:= max(m, j);
%p A373887   od;
%p A373887 m;
%p A373887 end proc:
%p A373887 map(f, [$1..100]);
%Y A373887 Cf. A001358, A096583, A373888.
%K A373887 nonn
%O A373887 1,2
%A A373887 _Robert Israel_, Aug 10 2024