cp's OEIS Frontend

A179746 Numbers of the form p^4*q^2*r^2 where p, q, and r are distinct primes.

%I A179746 #34 Apr 22 2025 04:16:04
%S A179746 3600,7056,8100,15876,17424,19600,22500,24336,39204,41616,48400,51984,
%T A179746 54756,67600,76176,86436,93636,94864,99225,115600,116964,121104,
%U A179746 122500,132496,138384,144400,171396,197136,211600,226576,240100,242064,245025
%N A179746 Numbers of the form p^4*q^2*r^2 where p, q, and r are distinct primes.
%C A179746 Numbers k such that tau(k^2)/tau(k) = 5 where tau(n) is the number of divisors of n (A000005). - _Bernard Schott_, Nov 27 2020
%H A179746 T. D. Noe, <a href="/A179746/b179746.txt">Table of n, a(n) for n = 1..1000</a>
%H A179746 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of Prime Signatures</a>
%H A179746 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A179746 Sum_{n>=1} 1/a(n) = (P(2)^2*P(4) - P(4)^2)/2 - P(2)*P(6) + P(8) = 0.00125114..., where P is the prime zeta function. - _Amiram Eldar_, Jul 03 2022
%F A179746 a(n) = A085987(n)^2. - _R. J. Mathar_, May 05 2023
%t A179746 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,2,4}; Select[Range[200000],f]
%o A179746 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\36)^(1/4), t1=p^4;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=q+1, sqrt(lim\t2), if(p==r,next);listput(v,t2*r^2)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 24 2011
%o A179746 (Python)
%o A179746 from math import isqrt
%o A179746 from sympy import primepi, primerange, integer_nthroot
%o A179746 def A179746(n):
%o A179746     def bisection(f,kmin=0,kmax=1):
%o A179746         while f(kmax) > kmax: kmax <<= 1
%o A179746         kmin = kmax >> 1
%o A179746         while kmax-kmin > 1:
%o A179746             kmid = kmax+kmin>>1
%o A179746             if f(kmid) <= kmid:
%o A179746                 kmax = kmid
%o A179746             else:
%o A179746                 kmin = kmid
%o A179746         return kmax
%o A179746     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**2)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))-primepi(integer_nthroot(x,4)[0])
%o A179746     return bisection(f,n,n)**2 # _Chai Wah Wu_, Mar 27 2025
%Y A179746 Subsequence of A217584.
%Y A179746 Cf. A189988 (tau(k^2)/tau(k) = 3).
%K A179746 nonn
%O A179746 1,1
%A A179746 _Vladimir Joseph Stephan Orlovsky_, Jul 25 2010