cp's OEIS Frontend

A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.

%I A066509 #66 Feb 28 2025 19:50:45
%S A066509 1309,1885,2013,2665,3729,5133,6061,6213,6305,6477,6853,6985,7257,
%T A066509 7953,8393,8533,8785,9213,9453,9821,9877,10281,10945,11605,12453,
%U A066509 12565,12801,12857,12993,13053,14133,14313,14329,14465,14817,15085,15265,15805,16113,16133
%N A066509 a(n) is the first of a triple of consecutive integers, each of which is both the product of three distinct primes and also the product of three primes counted with multiplicity.
%C A066509 A subsequence of A052214 and thus of A005238. - _M. F. Hasler_, Jan 05 2013
%C A066509 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
%C A066509 Minimal difference is 4 which occurs at indices n = {316, 547, 566, 604, 666, 695, 821, 874, 979, ...}. - _Zak Seidov_, Jul 04 2020
%H A066509 Amiram Eldar, <a href="/A066509/b066509.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%H A066509 G. L. Honaker, Jr. and Chris K. Caldwell, <a href="https://t5k.org/curios/page.php?short=1309">Prime Curios! 1309</a>.
%F A066509 a(n) == 1 (mod 4). - _Zak Seidov_, Mar 31 2020
%e A066509 a(5) = 3729 because it along with 3730 and 3731 are all the product of three distinct primes.
%t A066509 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 *)
%t A066509 SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==3,1,0],{n,17000}],{1,1,1}][[;;,1]] (* _Harvey P. Dale_, Feb 28 2025 *)
%o A066509 (PARI) Trip(n) = { local(f); f=factor(n); if (matsize(f)[1] != 3, return(0)); for(i=1, 3, if (f[i, 2] != 1, return(0))); return(1); } { n=0; for (m=1, 10^10, if (!Trip(m) || !Trip(m+1) || !Trip(m+2), next); write("b066509.txt", n++, " ", m); if (n==1000, return) ) } \\ _Harry J. Smith_, Feb 19 2010
%o A066509 (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
%o A066509 (PARI) is3dp(n)=my(f=factor(n));matsize(f)==[3,2]&&vecmax(f[,2])==1
%o A066509 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
%o A066509 (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
%Y A066509 Subsequence of A052214 and hence of A005238.
%Y A066509 Subsequence of A215217, A007675, A242606 and A168626.
%K A066509 nonn
%O A066509 1,1
%A A066509 _Jason Earls_, Jan 04 2002
%E A066509 Definition clarified by _Harvey P. Dale_, Feb 28 2025