A095678 Minimal sequence such that all triples of consecutive numbers have no common divisor greater than 1, but all three pairs within the triples are not coprime.

6, 10, 15, 12, 20, 45, 18, 40, 75, 24, 50, 135, 36, 80, 225, 48, 100, 375, 54, 160, 405, 72, 200, 675, 96, 250, 1125, 108, 320, 1215, 144, 400, 1875, 162, 500, 2025, 192, 640, 3375, 216, 800, 3645, 288, 1000, 5625, 324, 1250, 6075, 384, 1280, 9375, 432

Offset: 1

Author

Reinhard Zumkeller, Jul 04 2004

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A003586, A003592, A003593, A006530, A020639, A033845, A033846, A033849.

Programs

Mathematica

seq1[p_, q_, lim_] := Sort[Flatten[Table[p^i * q^j, {i, 1, Log[p, lim]}, {j, 1, Log[q, lim/p^i]}]]];
seq[lim_] := Module[{s1 = seq1[2, 3, lim], s2 = seq1[2, 5, lim], s3 = seq1[3, 5, lim], ns}, ns = Length[s3]; Flatten[Transpose[{s1[[1;;ns]], s2[[1;;ns]], s3}]]]; seq[10^4] (* Amiram Eldar, Sep 29 2024 *)

Formula

gcd(a(n),a(n+1),a(n+2)) = 1, gcd(a(n),a(n+1)) > 1, gcd(a(n),a(n+2)) > 1 and gcd(a(n+1),a(n+2)) > 1.

A001221(a(n)) = 2; 2 <= A020639(a(n)) <= 3 <= A006530(a(n)) <= 5.

From Jianing Song, Jun 08 2022: (Start)

a(3n-2) = A033845(n) = 6*A003586(n);

a(3n-1) = A033846(n) = 10*A003592(n);

a(3n) = A033849(n) = 15*A003593(n). (End)

Sum_{n>=1} 1/a(n) = 7/8. - Amiram Eldar, Sep 29 2024