cp's OEIS Frontend

A179698 Numbers of the form p^4*q^3*r where p, q, and r are distinct primes.

%I A179698 #20 Apr 22 2025 04:16:10
%S A179698 2160,3024,3240,4536,4752,5616,6000,7128,7344,8208,8424,9936,11016,
%T A179698 12312,12528,13392,14000,14904,15000,15984,16464,17712,18576,18792,
%U A179698 20088,20250,20304,22000,22896,23976,25488,26000,26352,26568,27440
%N A179698 Numbers of the form p^4*q^3*r where p, q, and r are distinct primes.
%H A179698 T. D. Noe, <a href="/A179698/b179698.txt">Table of n, a(n) for n = 1..1000</a>
%H A179698 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures/">List of Prime Signatures</a>
%H A179698 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%t A179698 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,3,4}; Select[Range[30000], f]
%o A179698 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\24)^(1/4), t1=p^4;forprime(q=2, (lim\t1)^(1/3), if(p==q, next);t2=t1*q^3;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 A179698 (Python)
%o A179698 from sympy import primepi, primerange, integer_nthroot
%o A179698 def A179698(n):
%o A179698     def bisection(f,kmin=0,kmax=1):
%o A179698         while f(kmax) > kmax: kmax <<= 1
%o A179698         kmin = kmax >> 1
%o A179698         while kmax-kmin > 1:
%o A179698             kmid = kmax+kmin>>1
%o A179698             if f(kmid) <= kmid:
%o A179698                 kmax = kmid
%o A179698             else:
%o A179698                 kmin = kmid
%o A179698         return kmax
%o A179698     def f(x): return n+x-sum(primepi(x//(p**4*q**3)) for p in primerange(integer_nthroot(x,4)[0]+1) for q in primerange(integer_nthroot(x//p**4,3)[0]+1))+sum(primepi(integer_nthroot(x//p**4,4)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[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 A179698     return bisection(f,n,n) # _Chai Wah Wu_, Mar 28 2025
%K A179698 nonn
%O A179698 1,1
%A A179698 _Vladimir Joseph Stephan Orlovsky_, Jul 24 2010