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A342950 7-smooth numbers not divisible by 10: positive numbers whose prime divisors are all <= 7 but do not contain both 2 and 5.

%I A342950 #34 Sep 18 2024 05:53:13
%S A342950 1,2,3,4,5,6,7,8,9,12,14,15,16,18,21,24,25,27,28,32,35,36,42,45,48,49,
%T A342950 54,56,63,64,72,75,81,84,96,98,105,108,112,125,126,128,135,144,147,
%U A342950 162,168,175,189,192,196,216,224,225,243,245,252,256,288,294,315,324
%N A342950 7-smooth numbers not divisible by 10: positive numbers whose prime divisors are all <= 7 but do not contain both 2 and 5.
%H A342950 David A. Corneth, <a href="/A342950/b342950.txt">Table of n, a(n) for n = 1..10195</a>
%F A342950 Sum_{n>=1} 1/a(n) = 63/16. - _Amiram Eldar_, Apr 01 2021
%e A342950 12 is in the sequence as all of its prime divisors are <= 7 and 12 is not divisible by 10.
%t A342950 Select[Range@500,Max[First/@FactorInteger@#]<=7&&Mod[#,10]!=0&] (* _Giorgos Kalogeropoulos_, Mar 30 2021 *)
%o A342950 (PARI) is(n) = if(n%10 == 0, return(0)); forprime(p = 2, 7, n/=p^valuation(n, p)); n==1
%o A342950 (Python)
%o A342950 A342950_list, n = [], 1
%o A342950 while n < 10**9:
%o A342950     if n % 10:
%o A342950         m = n
%o A342950         for p in (2,3,5,7):
%o A342950             q, r = divmod(m,p)
%o A342950             while r == 0:
%o A342950                 m = q
%o A342950                 q, r = divmod(m,p)
%o A342950         if m == 1:
%o A342950             A342950_list.append(n)
%o A342950     n += 1 # _Chai Wah Wu_, Mar 31 2021
%o A342950 (Python)
%o A342950 from sympy import integer_log
%o A342950 def A342950(n):
%o A342950     def bisection(f,kmin=0,kmax=1):
%o A342950         while f(kmax) > kmax: kmax <<= 1
%o A342950         while kmax-kmin > 1:
%o A342950             kmid = kmax+kmin>>1
%o A342950             if f(kmid) <= kmid:
%o A342950                 kmax = kmid
%o A342950             else:
%o A342950                 kmin = kmid
%o A342950         return kmax
%o A342950     def f(x):
%o A342950         c = n+x
%o A342950         for i in range(integer_log(x,7)[0]+1):
%o A342950             for j in range(integer_log(m:=x//7**i,3)[0]+1):
%o A342950                 c -= (k:=m//3**j).bit_length()+integer_log(k,5)[0]
%o A342950         return c
%o A342950     return bisection(f,n,n) # _Chai Wah Wu_, Sep 17 2024
%o A342950 (Python) # faster for initial segment of sequence
%o A342950 import heapq
%o A342950 from itertools import islice
%o A342950 def A342950gen(): # generator of terms
%o A342950     v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]
%o A342950     while True:
%o A342950         v = heapq.heappop(h)
%o A342950         if v != oldv:
%o A342950             yield v
%o A342950             oldv = v
%o A342950             for p in psmooth_primes:
%o A342950                 if not (p==2 and v%5==0) and not (p==5 and v&1==0):
%o A342950                     heapq.heappush(h, v*p)
%o A342950 print(list(islice(A342950gen(), 65))) # _Michael S. Branicky_, Sep 17 2024
%Y A342950 Union of A108319 and A108347.
%Y A342950 Intersection of A002473 and A067251.
%K A342950 nonn
%O A342950 1,2
%A A342950 _David A. Corneth_, Mar 30 2021