A361075 Products of exactly 7 distinct odd primes.

4849845, 5870865, 6561555, 7402395, 7912905, 8273265, 8580495, 8843835, 9444435, 10015005, 10140585, 10465455, 10555545, 10705695, 10818885, 10975965, 11565015, 11696685, 11996985, 12267255, 12777765, 12785955, 13096545, 13408395, 13498485, 13528515, 13667745, 13803405

Offset: 1

Author

Karl-Heinz Hofmann, Mar 01 2023

Keywords

• nonn

Examples

			a(1)     =   4849845 = 3*5*7*11*13*17*19
a(9663)  = 253808555 = 5*7*11*13*17*19*157
a(9961)  = 258573315 = 3*5*7*11*13*17*1013
a(10000) = 259173915 = 3*5*7*11*13*41*421
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A065091, A046388 (2 distinct odd primes).

Cf. A046389 (3 distinct odd primes), A046390 (4 distinct odd primes).

Cf. A046391 (5 distinct odd primes), A168352 (6 distinct odd primes).

Programs

Python

import numpy
from sympy import nextprime, sieve, primepi
k_upto = 14 * 10**6
array = numpy.zeros(k_upto,dtype="i4")
sieve_max_number = primepi(nextprime(k_upto // 255255))
for s in range(2,sieve_max_number):
    array[sieve[s]:k_upto][::sieve[s]] += 1
for s in range(2,sieve_max_number):
    array[sieve[s]**2:k_upto][::sieve[s]**2] = 0
print([k for k in range(1,k_upto,2) if array[k] == 7])

Python

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A361075(n):
    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)))
    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)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024