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A179689
Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.
8
1152, 3200, 6272, 8748, 15488, 21632, 36992, 46208, 54675, 67712, 107163, 107648, 123008, 175232, 215168, 236672, 264627, 282752, 312500, 359552, 369603, 445568, 476288, 574592, 632043, 645248, 682112, 703125, 789507, 798848, 881792, 1013888
OFFSET
1,1
FORMULA
Sum_{n>=1} 1/a(n) = P(2)*P(7) - P(9) = A085548 * A085967 - A085969 = 0.001741..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020
MAPLE
a:= proc(n) option remember; local k;
for k from 1+ `if` (n=1, 1, a(n-1))
while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 2]
do od; k
end:
seq (a(n), n=1..32); # Alois P. Heinz, Jan 23 2011
MATHEMATICA
f[n_]:=Sort[Last/@FactorInteger[n]]=={2, 7}; Select[Range[10^6], f]
PROG
(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim\4)^(1/7), t=p^7; forprime(q=2, sqrt(lim\t), if(p==q, next); listput(v, t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A179689(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return n+x-sum(primepi(isqrt(x//p**7)) for p in primerange(integer_nthroot(x, 7)[0]+1))+primepi(integer_nthroot(x, 9)[0])
return bisection(f, n, n) # Chai Wah Wu, Feb 21 2025
KEYWORD
nonn,changed
AUTHOR
EXTENSIONS
Title edited by Daniel Forgues, Jan 22 2011
STATUS
approved