A375511 a(n) is the common difference in the longest arithmetic progression of semiprimes ending in the n-th semiprime. If there is more than one such arithmetic progression, the smallest difference is chosen.

2, 3, 1, 4, 1, 6, 8, 3, 11, 12, 12, 1, 12, 1, 12, 14, 13, 17, 11, 12, 24, 7, 18, 35, 19, 24, 8, 24, 18, 29, 8, 36, 1, 24, 17, 30, 12, 4, 3, 48, 4, 36, 11, 48, 23, 24, 1, 30, 12, 13, 12, 36, 42, 24, 14, 16, 36, 14, 8, 32, 36, 7, 60, 42, 60, 60, 3, 4, 36, 46, 4, 12, 32, 4, 60, 16, 18, 44, 36, 16

Offset: 2

Author

Robert Israel, Aug 18 2024

Keywords

• nonn

Comments

a(n) is the smallest common difference in an arithmetic progression of A373887(n) semiprimes ending in A001358(n).

Examples

			The 5th semiprime is 14, A373887(5) = 3, and there are two arithmetic progressions of semiprimes of length 3 ending in 14, namely 6, 10, 14 with common difference 4 and 4, 9, 14 with common difference 5.  Therefore a(5) = min(4, 5) = 4.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001358, A373887, A375386.

Programs

Maple

S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):
f:= proc(n) local s, i, m, d, j, dm;
  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 dm fi;
    for j from 2 while ListTools:-BinarySearch(S, s-j*d) <> 0 do od;
    if j > m then m:= j; dm:= d fi;
  od;
dm;
end proc:
map(f, [$2..200]);