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

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, 54, 56, 63, 64, 72, 75, 81, 84, 96, 98, 105, 108, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 196, 216, 224, 225, 243, 245, 252, 256, 288, 294, 315, 324

Offset: 1

Author

David A. Corneth, Mar 30 2021

Keywords

• nonn

Examples

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

Links

• David A. Corneth, Table of n, a(n) for n = 1..10195

Crossrefs

Union of A108319 and A108347.

Intersection of A002473 and A067251.

Programs

Mathematica

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

PARI

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

Python

A342950_list, n = [], 1
while n < 10**9:
    if n % 10:
        m = n
        for p in (2,3,5,7):
            q, r = divmod(m,p)
            while r == 0:
                m = q
                q, r = divmod(m,p)
        if m == 1:
            A342950_list.append(n)
    n += 1 # Chai Wah Wu, Mar 31 2021

Python

from sympy import integer_log
def A342950(n):
    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
    def f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,3)[0]+1):
                c -= (k:=m//3**j).bit_length()+integer_log(k,5)[0]
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 17 2024

Python

# faster for initial segment of sequence
import heapq
from itertools import islice
def A342950gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                if not (p==2 and v%5==0) and not (p==5 and v&1==0):
                    heapq.heappush(h, v*p)
print(list(islice(A342950gen(), 65))) # Michael S. Branicky, Sep 17 2024

Formula

Sum_{n>=1} 1/a(n) = 63/16. - Amiram Eldar, Apr 01 2021