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

2,1

Comments

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

Links

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

Example

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.

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]);

Crossrefs

Cf. A001358, A373887, A375386.

Sequence in context: A235791 A035426 A065516 * A280699 A214967 A195164

Adjacent sequences: A375508 A375509 A375510 * A375512 A375513 A375514

Keyword

nonn

Author

Robert Israel, Aug 18 2024

Status

approved