cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A046310 Numbers that are divisible by exactly 8 primes counting multiplicity.

Original entry on oeis.org

256, 384, 576, 640, 864, 896, 960, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2176, 2240, 2400, 2432, 2496, 2916, 2944, 3024, 3136, 3168, 3240, 3264, 3360, 3520, 3600, 3648, 3712, 3744, 3968, 4000, 4160, 4374, 4416, 4536, 4704, 4736
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Comments

Also called 8-almost primes. Products of exactly 8 primes (not necessarily distinct). Any 8-almost prime can be represented in several ways as a product of two 4-almost primes A014613 and in several ways as a product of four semiprimes A001358. - Jonathan Vos Post, Dec 11 2004
Odd terms are in A046321; first odd term is a(64)=6561=3^8. - Zak Seidov, Feb 08 2016

Links

Crossrefs

Cf. A046309, A120049 (number of 8-almost primes <= 10^n).
Cf. A101637, A101638, A101605, A101606.
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), this sequence (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).
Cf. A046321.

Programs

  • Maple
    A046310 := proc(n)
    option remember;
    if n = 1 then
        2^8 ;
    else
        for a from procname(n-1)+1 do
            if numtheory[bigomega](a) = 8 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A046310(n),n=1..30) ; # R. J. Mathar, Dec 21 2018
  • Mathematica
    Select[Range[1600], Plus @@ Last /@ FactorInteger[ # ] == 8 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[5000],PrimeOmega[#]==8&]  (* Harvey P. Dale, Apr 19 2011 *)
  • PARI
    is(n)=bigomega(n)==8 \\ Charles R Greathouse IV, Mar 21 2013
  • Python
    from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046310(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,8)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Formula

Product p_i^e_i with Sum e_i = 8.
a(n) ~ 5040n log n / (log log n)^7. - Charles R Greathouse IV, May 06 2013
a(n) = A078840(8,n). - R. J. Mathar, Jan 30 2019

A046311 Numbers that are divisible by at least 9 primes (counted with multiplicity).

Original entry on oeis.org

512, 768, 1024, 1152, 1280, 1536, 1728, 1792, 1920, 2048, 2304, 2560, 2592, 2688, 2816, 2880, 3072, 3200, 3328, 3456, 3584, 3840, 3888, 4032, 4096, 4224, 4320, 4352, 4480, 4608, 4800, 4864, 4992, 5120, 5184, 5376, 5632, 5760, 5832, 5888, 6048, 6144
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Links

Crossrefs

Subsequence of A033987, A046304, A046305, A046307, and A046309.
Cf. A046312.

Programs

  • Mathematica
    Select[Range[6200],PrimeOmega[#]>8&] (* Harvey P. Dale, May 20 2013 *)
  • PARI
    is(n)=bigomega(n)>8 \\ Charles R Greathouse IV, Sep 17 2015
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046311(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+1+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,9)))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

Formula

Product p_i^e_i with Sum e_i >= 9.
a(n) = n + O(n (log log n)^7/log n). - Charles R Greathouse IV, Apr 07 2017

A046313 Numbers that are divisible by at least 10 primes (counted with multiplicity).

Original entry on oeis.org

1024, 1536, 2048, 2304, 2560, 3072, 3456, 3584, 3840, 4096, 4608, 5120, 5184, 5376, 5632, 5760, 6144, 6400, 6656, 6912, 7168, 7680, 7776, 8064, 8192, 8448, 8640, 8704, 8960, 9216, 9600, 9728, 9984, 10240, 10368, 10752, 11264, 11520, 11664, 11776
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Links

Crossrefs

Subsequence of A033987, A046304, A046305, A046307, A046309, and A046311.
Cf. A046314.

Programs

  • Mathematica
    Select[Range[12000],PrimeOmega[#]>9&] (* Harvey P. Dale, Dec 17 2018 *)
  • PARI
    is(n)=bigomega(n)>9 \\ Charles R Greathouse IV, Sep 17 2015
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046313(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,10)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Formula

Product p_i^e_i with Sum e_i >= 10.
a(n) = n + O(n (log log n)^8/log n). - Charles R Greathouse IV, Apr 07 2017
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.