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A361075 Products of exactly 7 distinct odd primes.

%I A361075 #14 Sep 10 2024 20:00:45
%S A361075 4849845,5870865,6561555,7402395,7912905,8273265,8580495,8843835,
%T A361075 9444435,10015005,10140585,10465455,10555545,10705695,10818885,
%U A361075 10975965,11565015,11696685,11996985,12267255,12777765,12785955,13096545,13408395,13498485,13528515,13667745,13803405
%N A361075 Products of exactly 7 distinct odd primes.
%H A361075 Karl-Heinz Hofmann, <a href="/A361075/b361075.txt">Table of n, a(n) for n = 1..10000</a>
%e A361075 a(1)     =   4849845 = 3*5*7*11*13*17*19
%e A361075 a(9663)  = 253808555 = 5*7*11*13*17*19*157
%e A361075 a(9961)  = 258573315 = 3*5*7*11*13*17*1013
%e A361075 a(10000) = 259173915 = 3*5*7*11*13*41*421
%o A361075 (Python)
%o A361075 import numpy
%o A361075 from sympy import nextprime, sieve, primepi
%o A361075 k_upto = 14 * 10**6
%o A361075 array = numpy.zeros(k_upto,dtype="i4")
%o A361075 sieve_max_number = primepi(nextprime(k_upto // 255255))
%o A361075 for s in range(2,sieve_max_number):
%o A361075     array[sieve[s]:k_upto][::sieve[s]] += 1
%o A361075 for s in range(2,sieve_max_number):
%o A361075     array[sieve[s]**2:k_upto][::sieve[s]**2] = 0
%o A361075 print([k for k in range(1,k_upto,2) if array[k] == 7])
%o A361075 (Python)
%o A361075 from math import prod, isqrt
%o A361075 from sympy import primerange, integer_nthroot, primepi
%o A361075 def A361075(n):
%o A361075     def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
%o A361075     def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,2,1,7)))
%o A361075     def bisection(f,kmin=0,kmax=1):
%o A361075         while f(kmax) > kmax: kmax <<= 1
%o A361075         while kmax-kmin > 1:
%o A361075             kmid = kmax+kmin>>1
%o A361075             if f(kmid) <= kmid:
%o A361075                 kmax = kmid
%o A361075             else:
%o A361075                 kmin = kmid
%o A361075         return kmax
%o A361075     return bisection(f,n,n) # _Chai Wah Wu_, Sep 10 2024
%Y A361075 Cf. A065091, A046388 (2 distinct odd primes).
%Y A361075 Cf. A046389 (3 distinct odd primes), A046390 (4 distinct odd primes).
%Y A361075 Cf. A046391 (5 distinct odd primes), A168352 (6 distinct odd primes).
%K A361075 nonn
%O A361075 1,1
%A A361075 _Karl-Heinz Hofmann_, Mar 01 2023