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.

A046312 Numbers that are divisible by exactly 9 primes with multiplicity.

Original entry on oeis.org

512, 768, 1152, 1280, 1728, 1792, 1920, 2592, 2688, 2816, 2880, 3200, 3328, 3888, 4032, 4224, 4320, 4352, 4480, 4800, 4864, 4992, 5832, 5888, 6048, 6272, 6336, 6480, 6528, 6720, 7040, 7200, 7296, 7424, 7488, 7936, 8000, 8320, 8748, 8832, 9072, 9408
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Comments

Also called 9-almost primes. Products of exactly 9 primes (not necessarily distinct). - Jonathan Vos Post, Dec 11 2004

Links

Crossrefs

Cf. A046311, A120050 (number of 9-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), A046310 (r = 8), this sequence (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). - Jason Kimberley, Oct 02 2011

Programs

  • Mathematica
    Select[Range[2200], Plus @@ Last /@ FactorInteger[ # ] == 9 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[10000],PrimeOmega[#]==9&] (* Harvey P. Dale, Oct 24 2020 *)
  • PARI
    is(n)=bigomega(n)==9 \\ Charles R Greathouse IV, Mar 21 2013
  • Python
    from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046312(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 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+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,9)))
    return bisection(f,n,n) # Chai Wah Wu, Nov 03 2024

Formula

Product p_i^e_i with Sum e_i = 9.
a(n) ~ 40320n log n / (log log n)^8. - Charles R Greathouse IV, May 06 2013

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

A294153 Numbers k = a * b, such that k' = a' * b' where k', a' and b' are the arithmetic derivatives of k, a and b.

Original entry on oeis.org

0, 1, 256, 512, 1152, 1728, 1920, 3072, 3456, 7776, 11664, 12800, 12960, 20736, 23328, 52488, 72000, 78732, 81920, 86400, 87480, 100352, 110208, 124800, 139968, 153216, 157464, 200000, 219520, 263424, 294912, 321024, 336000, 354294, 400000, 486000, 486720, 531441
Offset: 1

Views

Author

Paolo P. Lava, Oct 24 2017

Keywords

Comments

A046311 is a subset of this sequence.
Some numbers admit more than one couple of divisors a, b: 3456 = 8 * 432 = 54 * 64 and 3456' = 15552 = 8' * 432' = 12 * 1296 = 54' * 64' = 81 * 192.
Apart from the first term, squares of A165558 are part of the sequence. In A165558 n' = 2 * n and therefore (n^2)' = 2 * n * n' = 2 * n * 2 * n = (2 * n)^2. Thus n^2 = n * n and (n^2)' = n' * n'.

Examples

			a(0) = 0 because 0 = 0 * b and 0' = 0' * b' = 0;
a(1) = 1 because 1 = 1 * 1 and 1' = 1' * 1' = 0;
a(2) = 256 because 256 = 16 * 16 and 256' = 16' * 16' = 32 * 32 = 1024;
a(3) = 512 because 512 = 8 * 64 and 512' = 8' * 64' = 12 * 192 = 2304.

Links

Crossrefs

Cf. A003415, A046311, A165558.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,c,j,k,n,p;
for n from 1 to q do j:=sort([op(divisors(n))]);
for k from 2 to trunc((nops(j)+1)/2) do
a:=j[k]*add(op(2,p)/op(1,p), p=ifactors(j[k])[2]);
b:=(n/j[k])*add(op(2,p)/op(1,p), p=ifactors(n/j[k])[2]);
c:=n*add(op(2,p)/op(1,p), p=ifactors(n)[2]);
if c=a*b then print(n); break; fi; od; od; end: P(10^6);
Showing 1-3 of 3 results.

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