A342950 - OEIS

%I #34 Sep 18 2024 05:53:13

%S 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 54,56,63,64,72,75,81,84,96,98,105,108,112,125,126,128,135,144,147,

%U 162,168,175,189,192,196,216,224,225,243,245,252,256,288,294,315,324

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

%H David A. Corneth, <a href="/A342950/b342950.txt">Table of n, a(n) for n = 1..10195</a>

%F Sum_{n>=1} 1/a(n) = 63/16. - _Amiram Eldar_, Apr 01 2021

%e 12 is in the sequence as all of its prime divisors are <= 7 and 12 is not divisible by 10.

%t Select[Range@500,Max[First/@FactorInteger@#]<=7&&Mod[#,10]!=0&] (* _Giorgos Kalogeropoulos_, Mar 30 2021 *)

%o (PARI) is(n) = if(n%10 == 0, return(0)); forprime(p = 2, 7, n/=p^valuation(n, p)); n==1

%o (Python)

%o A342950_list, n = [], 1

%o while n < 10**9:

%o     if n % 10:

%o         m = n

%o         for p in (2,3,5,7):

%o             q, r = divmod(m,p)

%o             while r == 0:

%o                 m = q

%o                 q, r = divmod(m,p)

%o         if m == 1:

%o             A342950_list.append(n)

%o     n += 1 # _Chai Wah Wu_, Mar 31 2021

%o (Python)

%o from sympy import integer_log

%o def A342950(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     def f(x):

%o         c = n+x

%o         for i in range(integer_log(x,7)[0]+1):

%o             for j in range(integer_log(m:=x//7**i,3)[0]+1):

%o                 c -= (k:=m//3**j).bit_length()+integer_log(k,5)[0]

%o         return c

%o     return bisection(f,n,n) # _Chai Wah Wu_, Sep 17 2024

%o (Python) # faster for initial segment of sequence

%o import heapq

%o from itertools import islice

%o def A342950gen(): # generator of terms

%o     v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]

%o     while True:

%o         v = heapq.heappop(h)

%o         if v != oldv:

%o             yield v

%o             oldv = v

%o             for p in psmooth_primes:

%o                 if not (p==2 and v%5==0) and not (p==5 and v&1==0):

%o                     heapq.heappush(h, v*p)

%o print(list(islice(A342950gen(), 65))) # _Michael S. Branicky_, Sep 17 2024

%Y Union of A108319 and A108347.

%Y Intersection of A002473 and A067251.

%K nonn

%O 1,2

%A _David A. Corneth_, Mar 30 2021