cp's OEIS Frontend

A179691 Numbers p^5*q^2*r where p, q, r are 3 distinct primes.

%I A179691 #31 Apr 22 2025 04:12:50
%S A179691 1440,2016,2400,3168,3744,4704,4860,4896,5472,5600,6624,6804,7840,
%T A179691 8352,8800,8928,10400,10656,10692,11616,11808,12150,12384,12636,13536,
%U A179691 13600,15200,15264,16224,16524,16992,17248,17568,18400,18468,19296,19360
%N A179691 Numbers p^5*q^2*r where p, q, r are 3 distinct primes.
%H A179691 T. D. Noe, <a href="/A179691/b179691.txt">Table of n, a(n) for n = 1..1000</a>
%H A179691 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures/">List of Prime Signatures</a>
%H A179691 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%t A179691 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,5}; Select[Range[20000], f]
%o A179691 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/5), t1=p^5;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2011
%o A179691 (Python)
%o A179691 from math import isqrt
%o A179691 from sympy import primepi, primerange, integer_nthroot
%o A179691 def A179691(n):
%o A179691     def bisection(f,kmin=0,kmax=1):
%o A179691         while f(kmax) > kmax: kmax <<= 1
%o A179691         kmin = kmax >> 1
%o A179691         while kmax-kmin > 1:
%o A179691             kmid = kmax+kmin>>1
%o A179691             if f(kmid) <= kmid:
%o A179691                 kmax = kmid
%o A179691             else:
%o A179691                 kmin = kmid
%o A179691         return kmax
%o A179691     def f(x): return n+x-sum(primepi(x//(p**5*q**2)) for p in primerange(integer_nthroot(x,5)[0]+1) for q in primerange(isqrt(x//p**5)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(x//p**7) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,8)[0])<<1)
%o A179691     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A179691 Part of the list A178739 .. A179696 (and A030514 .. A030629, A189982 .. A189990 etc, cf. A101296). - _M. F. Hasler_, Jul 17 2019
%Y A179691 Subsequence of A175746 (numbers with 36 divisors).
%K A179691 nonn
%O A179691 1,1
%A A179691 _Vladimir Joseph Stephan Orlovsky_, Jul 24 2010
%E A179691 Name improved by _M. F. Hasler_, Jul 17 2019