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.

Showing 1-4 of 4 results.

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

A168628 Numbers n such that n and n+-1 have 4 distinct prime factors.

Original entry on oeis.org

37961, 44485, 45695, 50141, 51429, 55131, 55385, 61335, 63365, 64295, 67165, 68265, 68475, 70005, 70091, 71709, 72709, 76153, 80445, 81549, 81719, 82041, 84435, 85491, 86241, 90364, 95381, 97383, 98021, 99085, 99385, 99429, 99789, 100165, 100491, 100595
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

Keywords

Crossrefs

Subsequence of A140078.
Cf. A140077, A168626

Programs

  • Mathematica
    f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=4&&f[n-1]>=4&&f[n+1]>=4,AppendTo[lst,n]],{n,9!}];lst
SequencePosition[PrimeNu[Range[110000]],{4,4,4}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 27 2018 *)
  • PARI
    is(n)=omega(n)==4 && omega(n+1)==4 && omega(n-1)==4 \\ Charles R Greathouse IV, Jan 25 2025

Extensions

Corrected and extended by Harvey P. Dale, Apr 27 2018

A168629 Numbers n such that n,n+1 and sum of this two numbers have at least 3 distinct prime factors.

Original entry on oeis.org

1105, 1130, 1462, 1644, 1742, 1767, 2014, 2222, 2232, 2260, 2337, 2365, 2397, 2464, 2541, 2667, 2684, 2697, 2702, 2755, 2821, 2914, 3074, 3115, 3195, 3289, 3332, 3477, 3484, 3514, 3552, 3619, 3657, 3685, 3782, 3783, 3842, 3965, 4014, 4088, 4122, 4147, 4277
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

Keywords

Examples

			1105 = 5*13*17, 1106 = 2*7*79, 1105 + 1106 = 2211 = 3*11*67.

Crossrefs

Cf. A140077, A140078, A168626, A168628.

Programs

  • Maple
    q:= n-> andmap(x-> nops(numtheory[factorset](x))>2, [n, n+1, 2*n+1]):
select(q, [$1..4600])[];  # Alois P. Heinz, Jun 29 2021
  • Mathematica
    f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=3&&f[n+1]>=3&&f[n+n+1]>=3,AppendTo[lst,n]],{n,8!}];lst

A168630 Numbers n such that n, n+1, and the sum of those two numbers each have 4 or more distinct prime factors.

Original entry on oeis.org

46189, 50634, 69597, 76797, 90117, 97954, 108205, 115804, 127347, 138957, 144627, 159340, 164020, 166022, 166497, 166705, 167205, 167485, 173194, 174454, 181670, 186294, 190014, 193154, 198789, 211029, 212134, 214225, 217217, 221815, 222547, 224146
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

Keywords

Examples

			FactorInteger[46189]=11*13*17*19, FactorInteger[46190]=2*5*31*149, FactorInteger[46189+46190]=3*7*53*83,..

Links

Crossrefs

Cf. A140077, A140078, A168626, A168628, A168629

Programs

  • Maple
    g:= proc(n) option remember; nops(numtheory:-factorset(n))>=4 end proc:
filter:= n -> g(n) and g(n+1) and g(2*n+1):
select(filter, [$1..300000]); # Robert Israel, May 09 2018
  • Mathematica
    f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=4&&f[n+1]>=4&&f[n+n+1]>=4,AppendTo[lst,n]],{n,9!}];lst
Select[Range[225000],Min[Thread[PrimeNu[{#,#+1,2#+1}]]]>3&](* Harvey P. Dale, Nov 11 2017 *)

Extensions

Definition modified and terms extended by Harvey P. Dale, Nov 11 2017
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.