A373887 - OEIS

%I #38 Aug 21 2024 05:40:17

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

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

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

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

%C 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 Robert Israel, <a href="/A373887/b373887.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):

%p f:= proc(n) local s,i,m,d,j;

%p   m:= 1;

%p   s:= S[n];

%p   for i from n-1 to 1 by -1 do

%p     d:= s - S[i];

%p     if s - m*d < 4 then return m fi;

%p     for j from 2 while ListTools:-BinarySearch(S,s-j*d) <> 0 do od;

%p     m:= max(m, j);

%p   od;

%p m;

%p end proc:

%p map(f, [$1..100]);

%Y Cf. A001358, A096583, A373888.

%K nonn

%O 1,2

%A _Robert Israel_, Aug 10 2024