cp's OEIS Frontend

A143610 Numbers of the form p^2 * q^3, where p,q are distinct primes.

%I A143610 #74 Feb 21 2025 18:06:19
%S A143610 72,108,200,392,500,675,968,1125,1323,1352,1372,2312,2888,3087,3267,
%T A143610 4232,4563,5324,6125,6728,7688,7803,8575,8788,9747,10952,11979,13448,
%U A143610 14283,14792,15125,17672,19652,19773,21125,22472,22707,25947,27436
%N A143610 Numbers of the form p^2 * q^3, where p,q are distinct primes.
%C A143610 Also: numbers with prime signature {3,2}.
%C A143610 This is a subsequence of A114128. [Hasler]
%C A143610 Every a(n) is an Achilles number (A052486). They are minimal, meaning no proper divisor is an Achilles number. - _Antonio Roldán_, Dec 27 2011
%H A143610 T. D. Noe, <a href="/A143610/b143610.txt">Table of n, a(n) for n = 1..1000</a>
%H A143610 Project Euler, <a href="https://projecteuler.net/problem=200">Problem 200: Find the 200th prime-proof sqube containing the contiguous sub-string 200</a>.
%H A143610 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A143610 Sum_{n>=1} 1/a(n) = P(2)*P(3) - P(5) = A085548 * A085541 - A085965 = 0.043280..., where P is the prime zeta function. - _Amiram Eldar_, Jul 06 2020
%e A143610 The first three terms of this sequence are 3^2 * 2^3 = 72, 2^2 * 3^3 = 108, 5^2 * 2^3 = 200.
%t A143610 f[n_] := Sort[Last/@FactorInteger[n]] == {2, 3}; Select[Range[30000], f] (* _Vladimir Joseph Stephan Orlovsky_, Oct 09 2009 *)
%o A143610 (PARI) for(n=1, 10^5, omega(n)==2 || next; vecsort(factor(n)[,2])==[2,3]~ && print1(n","))
%o A143610 (PARI) list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/3), t=p^3;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A143610 (Python)
%o A143610 from math import isqrt
%o A143610 from sympy import primepi, primerange, integer_nthroot
%o A143610 def A143610(n):
%o A143610     def bisection(f,kmin=0,kmax=1):
%o A143610         while f(kmax) > kmax: kmax <<= 1
%o A143610         kmin = kmax >> 1
%o A143610         while kmax-kmin > 1:
%o A143610             kmid = kmax+kmin>>1
%o A143610             if f(kmid) <= kmid:
%o A143610                 kmax = kmid
%o A143610             else:
%o A143610                 kmin = kmid
%o A143610         return kmax
%o A143610     def f(x): return n+x-sum(primepi(isqrt(x//p**3)) for p in primerange(integer_nthroot(x,3)[0]+1))+primepi(integer_nthroot(x,5)[0])
%o A143610     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A143610 Cf. A114128.
%Y A143610 Cf. A085541, A085548, A085965.
%K A143610 easy,nonn
%O A143610 1,1
%A A143610 _M. F. Hasler_, Aug 27 2008