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

1,2

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Union of A108319 and A108347.

Intersection of A002473 and A067251.

Sequence in context: A050742 A350572 A290387 * A238985 A337230 A226900

Adjacent sequences: A342947 A342948 A342949 * A342951 A342952 A342953

Keyword

nonn

Author

David A. Corneth, Mar 30 2021

Status

approved