A066509 a(n) is the first of a triple of consecutive integers, each of which is the product of three distinct primes.

1309, 1885, 2013, 2665, 3729, 5133, 6061, 6213, 6305, 6477, 6853, 6985, 7257, 7953, 8393, 8533, 8785, 9213, 9453, 9821, 9877, 10281, 10945, 11605, 12453, 12565, 12801, 12857, 12993, 13053, 14133, 14313, 14329, 14465, 14817, 15085, 15265, 15805, 16113, 16133

Offset

1,1

Comments

A subsequence of A052214 and thus of A005238. - M. F. Hasler, Jan 05 2013

Also, the start of pairs of adjacent sphenic twins, i.e., a(n) = A215217(k) such that A215217(k+1) = A215217(k)+1. Therefore these triples might be called "sphenic triples". They form a subsequence of A242606. - M. F. Hasler, May 18 2014

Minimal difference is 4 which occurs at indices n = {316, 547, 566, 604, 666, 695, 821, 874, 979, ...}. - Zak Seidov, Jul 04 2020

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 1309.

Formula

a(n) == 1 (mod 4). - Zak Seidov, Mar 31 2020

Example

a(5) = 3729 because it along with 3730 and 3731 are all the product of three distinct primes.

Mathematica

f[n_]:=Last/@FactorInteger[n]=={1, 1, 1}; lst={}; Do[If[f[n]&&f[n+1]&&f[n+2], AppendTo[lst, n]], {n, 9!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 04 2010 *)
SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3, 1, 0], {n, 17000}], {1, 1, 1}][[;; , 1]] (* Harvey P. Dale, Feb 28 2025 *)

Prog

(PARI) A066509(n, show_all=0, a=2*3*5, s=[1, 1, 1]~)={until( !n-- || !a++, until(, factor(a+2)[, 2]!=s && (a+=3) && next; factor(a+1)[, 2]!=s && (a+=2) && next; factor(a)[, 2]==s && break; factor(a+3)[, 2]==s && a++ && break; a+=4); show_all && print1(a", ")); a} \\ M. F. Hasler, Jan 05 2013
(PARI) is3dp(n)=my(f=factor(n)); matsize(f)==[3, 2]&&vecmax(f[, 2])==1
list(lim)=my(v=List(), t); forprime(p=17, lim\15, forprime(q=5, min(p-1, lim\3), forprime(r=3, min(q-1, lim\(p*q)), t=p*q*r; if(t%4==1 && is3dp(t+1) && is3dp(t+2), listput(v, t))))); Set(v) \\ Charles R Greathouse IV, Jan 05 2013; updated Jan 22 2025
(PARI) list(lim)=my(v=List(), ct); forfactored(n=1309, lim\1+2, if(n[2][, 2]==[1, 1, 1]~, if(ct++==3, listput(v, n[1]-2)), ct=0)); Vec(v) \\ Charles R Greathouse IV, Aug 30 2022

Crossrefs

Subsequence of A052214 and hence of A005238.

Subsequence of A215217, A007304, A007675, A242606 and A168626.

Sequence in context: A209853 A165936 A242606 * A248202 A256668 A205301

Adjacent sequences: A066506 A066507 A066508 * A066510 A066511 A066512

Keyword

nonn

Author

Jason Earls, Jan 04 2002

Extensions

Definition clarified by Harvey P. Dale, Feb 28 2025

Status

approved