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A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7*q*r with p, q and r distinct primes.

%I A179696 #34 Apr 22 2025 04:16:13
%S A179696 1920,2688,4224,4480,4992,6528,7040,7296,8320,8832,9856,10880,11136,
%T A179696 11648,11904,12160,14208,14720,15232,15744,16512,17024,18048,18304,
%U A179696 18560,19840,20352,20608,21870,22656,23424,23680,23936,25728,25984,26240,26752,27264
%N A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7*q*r with p, q and r distinct primes.
%H A179696 T. D. Noe, <a href="/A179696/b179696.txt">Table of n, a(n) for n = 1..1000</a>
%H A179696 Will Nicholes <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures/">List of Prime Signatures</a>
%H A179696 OEIS Wiki, <a href="http://oeis.org/wiki/Prime_signature#Numbers_with_same_prime_signature">Numbers with same prime signature</a>.
%p A179696 a:= proc(n) option remember; local k;
%p A179696       for k from 1+ `if` (n=1, 1, a(n-1))
%p A179696         while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 1, 1]
%p A179696       do od; k
%p A179696     end:
%p A179696 seq (a(n), n=1..40); # _Alois P. Heinz_, Jan 23 2011
%t A179696 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,7}; Select[Range[30000], f]
%o A179696 (PARI) list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\6)^(1/7), t1=p^7;forprime(q=2, lim\t1, if(p==q, next);t2=t1*q;forprime(r=q+1, lim\t2, if(p==r,next);listput(v,t2*r)))); vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 20 2011
%o A179696 (Python)
%o A179696 from math import isqrt
%o A179696 from sympy import primerange, primepi, integer_nthroot
%o A179696 def A179696(n):
%o A179696     def bisection(f,kmin=0,kmax=1):
%o A179696         while f(kmax) > kmax: kmax <<= 1
%o A179696         kmin = kmax >> 1
%o A179696         while kmax-kmin > 1:
%o A179696             kmid = kmax+kmin>>1
%o A179696             if f(kmid) <= kmid:
%o A179696                 kmax = kmid
%o A179696             else:
%o A179696                 kmin = kmid
%o A179696         return kmax
%o A179696     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**7)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,7)[0]+1))+sum(primepi(x//p**8) for p in primerange(integer_nthroot(x,8)[0]+1))-primepi(integer_nthroot(x,9)[0])
%o A179696     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A179696 Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695.
%K A179696 nonn
%O A179696 1,1
%A A179696 _Vladimir Joseph Stephan Orlovsky_, Jul 24 2010
%E A179696 Title edited by _Daniel Forgues_, Jan 22 2011