A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

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

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

OEIS Wiki, Numbers with same prime signature.

Will Nicholes, Prime Signatures

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

Crossrefs

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688.

Cf. A085548, A085967, A085969.

Sequence in context: A114046 A035888 A252438 * A131527 A229590 A269216

Adjacent sequences: A179686 A179687 A179688 * A179690 A179691 A179692

Keyword

nonn,changed

Author

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

Extensions

Title edited by Daniel Forgues, Jan 22 2011

Status

approved