A080194 7-smooth numbers which are not 5-smooth.

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 84, 98, 105, 112, 126, 140, 147, 168, 175, 189, 196, 210, 224, 245, 252, 280, 294, 315, 336, 343, 350, 378, 392, 420, 441, 448, 490, 504, 525, 560, 567, 588, 630, 672, 686, 700, 735, 756, 784, 840, 875, 882, 896, 945, 980

Offset

1,1

Comments

Numbers of the form 7*2^r*3^s*5^t*7^u with r, s, t, u >= 0.

Multiples of 7 which are members of A002473. Or multiples of 7 with the largest prime divisor < 10.

Numbers whose greatest prime factor (A006530) is 7. - M. F. Hasler, Nov 21 2018

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Formula

a(n) = 7 * A002473(n). - David A. Corneth, Nov 22 2018

Sum_{n>=1} 1/a(n) = 5/8. - Amiram Eldar, Nov 10 2020

Example

28 = 2^2*7 is a term but 30 = 2*3*5 is not.

Mathematica

Select[Range[999], FactorInteger[#][[-1, 1]] == 7 &] (* Giovanni Resta, Nov 22 2018 *)

Prog

(PARI) A080194_list(M)={my(L=List(), a, b, c); for(r=1, logint(M\1, 7), a=7^r; for(s=0, logint(M\a, 3), b=a*3^s; for(t=0, logint(M\b, 5), c=b*5^t; for(u=0, logint(M\c, 2), listput(L, c<<u))))); Set(L)} \\ Could be replaced by smooth(primes(4), M) from A051037. - Edited by M. F. Hasler, Nov 22 2018
(PARI) select( is_A080194(n)={n>1 && vecmax(factor(n, 7)[, 1])==7}, [0..10^3]) \\ Defines is_A080194(), used elsewhere. The select() command is a check and illustration. For longer lists, use list() above. - M. F. Hasler, Nov 21 2018
(Python)
from sympy import integer_log
def A080194(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, 5)[0]+1):
                for k in range(integer_log(r:=m//5**j, 3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f, n, n)*7 # Chai Wah Wu, Sep 17 2024
(Python) # faster for initial segment of sequence
import heapq
from itertools import islice
from sympy import primerange
def A080194gen(p=7): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield 7*v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A080194gen(), 65))) # Michael S. Branicky, Sep 17 2024

Crossrefs

Cf. A002473, A051037.

Cf. A085125, A085126, A085127, A085128, A085129, A085131, A085132.

Sequence in context: A182341 A008589 A085130 * A206717 A043393 A028437

Adjacent sequences: A080191 A080192 A080193 * A080195 A080196 A080197

Keyword

easy,nonn

Author

Klaus Brockhaus, Feb 10 2003

Status

approved