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 1-5 of 5 results.

A046306 Numbers that are divisible by exactly 6 primes with multiplicity.

Original entry on oeis.org

64, 96, 144, 160, 216, 224, 240, 324, 336, 352, 360, 400, 416, 486, 504, 528, 540, 544, 560, 600, 608, 624, 729, 736, 756, 784, 792, 810, 816, 840, 880, 900, 912, 928, 936, 992, 1000, 1040, 1104, 1134, 1176, 1184, 1188, 1215, 1224, 1232, 1260, 1312, 1320
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Comments

Also called 6-almost primes. Products of exactly 6 primes (not necessarily distinct). Any 6-almost prime can be represented in several ways as a product of two 3-almost primes A014612 and in several ways as a product of three semiprimes A001358. - Jonathan Vos Post, Dec 11 2004

Links

Crossrefs

Cf. A046305, A120047 (number of 6-almost primes <= 10^n).
Cf. 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), this sequence (r = 6), A046308 (r = 7), A046310 (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).

Programs

  • Mathematica
    Select[Range[500], Plus @@ Last /@ FactorInteger[ # ] == 6 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[1400],PrimeOmega[#]==6&] (* Harvey P. Dale, May 21 2012 *)
  • PARI
    is(n)=bigomega(n)==6 \\ Charles R Greathouse IV, Mar 21 2013
  • Python
    from math import isqrt, prod
from sympy import primepi, primerange, integer_nthroot
def A046306(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,0,1,1,6)))
    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 = 6.
a(n) ~ 120n log n / (log log n)^5. - Charles R Greathouse IV, May 06 2013
a(n) = A078840(6,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

A166719 Numbers with at most 5 prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73
Offset: 1

Views

Author

Michael B. Porter, Oct 20 2009

Keywords

Comments

Complement of A046305, A001222(a(n))<=5

Examples

			50 = 2*5*5 is in the sequence since it has 3 prime factors and 3 <= 5
64 = 2*2*2*2*2*2 is not in the sequence since it has 6 prime factors

Links

Crossrefs

Cf. A046305, A001222
For numbers with at most n prime factors: n=1: A000040, n=2: A037143, n=3: A037144, n=4: A166718.

Programs

  • Mathematica
    Select[Range[100],PrimeOmega[#]<6&] (* Harvey P. Dale, Jul 13 2011 *)
  • PARI
    isA166719(n) = (bigomega(n) <= 5)

A346207 Numbers k such that k and k+1 are products of at least 6 primes.

Original entry on oeis.org

1215, 3968, 5103, 5264, 6560, 7424, 7695, 8991, 9375, 9800, 11024, 11583, 11744, 12375, 12879, 13040, 14175, 14336, 14624, 15624, 16064, 16280, 16767, 16928, 17199, 17576, 18224, 21375, 21735, 22112, 22599, 22815, 23408, 24255, 24543, 24704, 24975, 25839, 26000, 26487, 27135
Offset: 1

Views

Author

Tanya Khovanova, Jul 10 2021

Keywords

Comments

Integers k such that k and k+1 are in A046305.

Examples

			1215 = 3^5*5 is a product of 6 primes. The next integer, 1216 = 2^6*19, is a product of 7 primes. Thus, 1215 is in this sequence.

Crossrefs

Cf. A046305, A345744.

Programs

  • Maple
    q:= n-> andmap(x-> numtheory[bigomega](x)>5, [n, n+1]):
select(q, [$1..30000])[];  # Alois P. Heinz, Jul 10 2021
  • Mathematica
    Select[Range[100000], Total[Transpose[FactorInteger[#]][[2]]] >= 6 && Total[Transpose[FactorInteger[# + 1]][[2]]] >= 6 &]
  • PARI
    isA346207(k) = (bigomega(k) >= 6) && (bigomega(k+1) >= 6) \\ Jianing Song, Jul 10 2021
  • Python
    from sympy import factorint
def prod6(n): return sum(factorint(n).values()) >= 6
def aupto(lim): return [k for k in range(lim+1) if prod6(k) and prod6(k+1)]
print(aupto(27135)) # Michael S. Branicky, Jul 10 2021
Showing 1-5 of 5 results.

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