login
Numbers that are divisible by exactly 5 different primes.
26

%I #57 Apr 22 2024 09:37:54

%S 2310,2730,3570,3990,4290,4620,4830,5460,5610,6006,6090,6270,6510,

%T 6630,6930,7140,7410,7590,7770,7854,7980,8190,8580,8610,8778,8970,

%U 9030,9240,9282,9570,9660,9690,9870,10010,10230,10374,10626,10710,10920,11130,11220,11310

%N Numbers that are divisible by exactly 5 different primes.

%H T. D. Noe, <a href="/A051270/b051270.txt">Table of n, a(n) for n = 1..1000</a>

%H Hans Montanus and Ron Westdijk, <a href="https://greenbluemath.nl/wp-content/uploads/2024/03/Cellular-Automation-and-Binomials.pdf">Cellular Automation and Binomials</a>, Green Blue Mathematics (2022), p. 90.

%e 2730 = 2*3*5*7*13 is the first nontrivial 5-prime factor number following the 5th primorial, 2310 = 2*3*5*7*11.

%p A051270 := proc(n)

%p option remember;

%p local a;

%p if n = 1 then

%p 2*3*5*7*11 ;

%p else

%p for a from procname(n-1)+1 do

%p if A001221(a)= 5 then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc: # _R. J. Mathar_, Oct 13 2019

%t Select[Range[12000],PrimeNu[#]==5&] (* _Harvey P. Dale_, Feb 13 2012 *)

%o (PARI) is(n)=omega(n)==5 \\ _Charles R Greathouse IV_, Apr 29 2015

%o (PARI) A246655(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v)

%o list(lim,pr=5)=if(pr==1, return(A246655(lim))); my(v=List(),pr1=pr-1,mx=prod(i=1,pr1,prime(i))); forprime(p=prime(pr),lim\mx, my(u=list(lim\p,pr1)); for(i=1,#u,listput(v,p*u[i]))); Set(v) \\ _Charles R Greathouse IV_, Feb 03 2023

%o (Python)

%o from sympy import primefactors

%o print([n for n in range(2, 20001) if len(primefactors(n))==5]) # _Indranil Ghosh_, Apr 06 2017

%Y A046303 is a subsequence.

%Y Row 5 of A125666.

%Y Cf. A000977, A002110.

%K nonn

%O 1,1

%A _Labos Elemer_