cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Jason Earls, Jan 04 2002

Keywords

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

Examples

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

Links

Crossrefs

Subsequence of A052214 and hence of A005238.
Subsequence of A215217, A007675, A242606 and A168626.

Programs

  • 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 *)
  • 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
  • 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

Formula

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

Extensions

Definition clarified by Harvey P. Dale, Feb 28 2025

A242605 Start of a triple of consecutive squarefree numbers which are all semiprimes.

Original entry on oeis.org

33, 55, 85, 91, 93, 115, 118, 119, 141, 142, 143, 158, 201, 202, 203, 205, 213, 214, 215, 217, 218, 295, 298, 299, 301, 302, 323, 326, 391, 393, 411, 413, 445, 451, 511, 514, 535, 542, 551, 622, 633, 685, 694, 695, 697, 745, 763, 778, 791, 799, 815, 842, 843, 865, 898, 921, 922
Offset: 1

Views

Author

M. F. Hasler, May 18 2014

Keywords

Comments

Sequence A039833 is a subsequence.

Examples

			33 is in the sequence because 33, 34, 35 are all squarefree semiprimes.
55 is in the sequence because 55, 57, 58 (we ignore 56 because it's not squarefree) are all squarefree semiprimes.

Links

Crossrefs

Cf. A242606 (m=3), A242607 (m=4), A242608 (m=5), A242621 (first terms for positive m).

Programs

  • Mathematica
    Transpose[Select[Partition[Select[Range[1000],SquareFreeQ],3,1], Union[ PrimeOmega[ #]] =={2}&]][[1]] (* Harvey P. Dale, Feb 07 2016 *)
  • PARI
    is_A242605(n,c=2)==issquarefree(n)&&omega(n)==2&&(!c||until(issquarefree(n++),)||is_A242605(n,c-1))
  • PARI
    (back(n,c=1)=until(issquarefree(n--)&&c--,);n); for(n=1,999,issquarefree(n)||next;dk==4&&dk==dm&&numdiv(n)==dm&&print1(back(n)",");dk=dm;dm=numdiv(n))
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