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A201266 The seventh divisor of numbers with exactly 49 divisors.

%I A201266 #12 Feb 22 2025 20:04:04
%S A201266 9,16,16,27,49,22,26,81,32,125,32,81,32,81,125,81,32,32,169,81,37,343,
%T A201266 41,289,43,87,343,93,47,361,53,111,529,59,343,61,123,129,361,64,141,
%U A201266 64,1331,625,64,625,64,159,529,64,177,64,183,625,1331,64,201,64
%N A201266 The seventh divisor of numbers with exactly 49 divisors.
%H A201266 Reinhard Zumkeller, <a href="/A201266/b201266.txt">Table of n, a(n) for n = 1..1000</a>
%e A201266 a(1) = A114334(7);
%e A201266 a(2) = A159765(7).
%o A201266 (Haskell)
%o A201266 a201266 n = [d | d <- [1..], a175755 n `mod` d == 0] !! 6
%o A201266 (Python)
%o A201266 from math import isqrt
%o A201266 from sympy import primepi, integer_nthroot, primerange, divisors
%o A201266 def A201266(n):
%o A201266     def bisection(f,kmin=0,kmax=1):
%o A201266         while f(kmax) > kmax: kmax <<= 1
%o A201266         kmin = kmax >> 1
%o A201266         while kmax-kmin > 1:
%o A201266             kmid = kmax+kmin>>1
%o A201266             if f(kmid) <= kmid:
%o A201266                 kmax = kmid
%o A201266             else:
%o A201266                 kmin = kmid
%o A201266         return kmax
%o A201266     def f(x): return int(n+x+(t:=primepi(s:=isqrt(y:=integer_nthroot(x,6)[0])))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1))-primepi(integer_nthroot(x,48)[0]))
%o A201266     return divisors(bisection(f,n,n))[6] # _Chai Wah Wu_, Feb 22 2025
%Y A201266 Cf. A175755, A135581.
%K A201266 nonn
%O A201266 1,1
%A A201266 _Reinhard Zumkeller_, Nov 29 2011