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A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

%I A179689 #33 Feb 21 2025 19:39:02
%S A179689 1152,3200,6272,8748,15488,21632,36992,46208,54675,67712,107163,
%T A179689 107648,123008,175232,215168,236672,264627,282752,312500,359552,
%U A179689 369603,445568,476288,574592,632043,645248,682112,703125,789507,798848,881792,1013888
%N A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.
%H A179689 T. D. Noe, <a href="/A179689/b179689.txt">Table of n, a(n) for n = 1..1000</a>
%H A179689 OEIS Wiki, <a href="http://oeis.org/wiki/Prime_signature#Numbers_with_same_prime_signature">Numbers with same prime signature</a>.
%H A179689 Will Nicholes, <a href="http://willnicholes.com/math/primesiglist.htm">Prime Signatures</a>
%F A179689 Sum_{n>=1} 1/a(n) = P(2)*P(7) - P(9) = A085548 * A085967 - A085969 = 0.001741..., where P is the prime zeta function.  - _Amiram Eldar_, Jul 06 2020
%p A179689 a:= proc(n) option remember; local k;
%p A179689       for k from 1+ `if` (n=1, 1, a(n-1))
%p A179689         while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 2]
%p A179689       do od; k
%p A179689     end:
%p A179689 seq (a(n), n=1..32);  # _Alois P. Heinz_, Jan 23 2011
%t A179689 f[n_]:=Sort[Last/@FactorInteger[n]]=={2,7}; Select[Range[10^6], f]
%o A179689 (PARI) list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/7), t=p^7;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 A179689 (Python)
%o A179689 from math import isqrt
%o A179689 from sympy import primepi, integer_nthroot, primerange
%o A179689 def A179689(n):
%o A179689     def bisection(f,kmin=0,kmax=1):
%o A179689         while f(kmax) > kmax: kmax <<= 1
%o A179689         kmin = kmax >> 1
%o A179689         while kmax-kmin > 1:
%o A179689             kmid = kmax+kmin>>1
%o A179689             if f(kmid) <= kmid:
%o A179689                 kmax = kmid
%o A179689             else:
%o A179689                 kmin = kmid
%o A179689         return kmax
%o A179689     def f(x): return n+x-sum(primepi(isqrt(x//p**7)) for p in primerange(integer_nthroot(x,7)[0]+1))+primepi(integer_nthroot(x,9)[0])
%o A179689     return bisection(f,n,n) # _Chai Wah Wu_, Feb 21 2025
%Y A179689 Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688.
%Y A179689 Cf. A085548, A085967, A085969.
%K A179689 nonn
%O A179689 1,1
%A A179689 _Vladimir Joseph Stephan Orlovsky_, Jul 24 2010
%E A179689 Title edited by _Daniel Forgues_, Jan 22 2011