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 1-6 of 6 results.

A014614 Numbers that are products of 5 primes (or 5-almost primes, a generalization of semiprimes).

Original entry on oeis.org

32, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, 200, 208, 243, 252, 264, 270, 272, 280, 300, 304, 312, 368, 378, 392, 396, 405, 408, 420, 440, 450, 456, 464, 468, 496, 500, 520, 552, 567, 588, 592, 594, 612, 616, 630, 656, 660, 675, 680, 684, 688, 696
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Divisible by exactly 5 prime powers (not including 1).

Links

Crossrefs

Cf. A046304, A114453 (number of 5-almost primes <= 10^n).
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), this sequence (r = 5), A046306 (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). - Jason Kimberley, Oct 02 2011

Programs

  • Mathematica
    Select[Range[300], Plus @@ Last /@ FactorInteger[ # ] == 5 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
  • PARI
    is(n)=bigomega(n)==5 \\ Charles R Greathouse IV, Mar 20 2013
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A014614(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m*r*s))-d for a,k in enumerate(primerange(integer_nthroot(x,5)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,4)[0]+1),a) for c,r in enumerate(primerange(m,integer_nthroot(x//(k*m),3)[0]+1),b) for d,s in enumerate(primerange(r,isqrt(x//(k*m*r))+1),c)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024

Formula

Product p_i^e_i with sum e_i = 5.
a(n) ~ 24n log n/(log log n)^4. - Charles R Greathouse IV, Mar 20 2013
a(n) = A078840(5,n). - R. J. Mathar, Jan 30 2019

Extensions

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu) and Patrick De Geest, Jun 15 1998

A046305 Numbers that are divisible by at least 6 primes (counted with multiplicity).

Original entry on oeis.org

64, 96, 128, 144, 160, 192, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 544, 560, 576, 600, 608, 624, 640, 648, 672, 704, 720, 729, 736, 756, 768, 784, 792, 800, 810, 816, 832, 840, 864, 880, 896
Offset: 1

Views

Author

Patrick De Geest, Jun 15 1998

Keywords

Links

Crossrefs

Subsequence of A033987 and A046304.
Cf. A046306.

Programs

  • Mathematica
    Select[Range[1000],Total[Transpose[FactorInteger[#]][[2]]]>5&]  (* Harvey P. Dale, Jan 13 2011 *)
Select[Range[1000],PrimeOmega[#]>5&] (* Harvey P. Dale, Apr 14 2019 *)
  • PARI
    is(n)=bigomega(n)>5 \\ Charles R Greathouse IV, Sep 17 2015
  • Python
    from math import prod, isqrt
from sympy import primerange, primepi, integer_nthroot
def A046305(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 almostprimepi(n, k):
        if k==0: return int(n>=1)
        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)))
        return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
    def f(x): return n+1+sum(almostprimepi(x, k) for k in range(1, 6))
    return bisection(f, n, n) # Chai Wah Wu, Mar 29 2025

Formula

Product p_i^e_i with Sum e_i >= 6.
a(n) = n + O(n (log log n)^4/log n). - Charles R Greathouse IV, Apr 07 2017

Extensions

Offset corrected by Andrew Howroyd, Aug 13 2024

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

A166718 Numbers with at most 4 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, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76
Offset: 1

Views

Author

Michael B. Porter, Oct 20 2009

Keywords

Comments

Complement of A046304, A001222(a(n)) <= 4.
Maynard shows there are infinitely many integers n such that the interval [n,n+90] contains 2 primes and a number with at most 4 prime factors [Jonathan Vos Post, May 23 2012]
Subset of the 5-free numbers (numbers where each exponent in the prime factorization is <=4). - R. J. Mathar, Aug 08 2012

Examples

			88 = 2*2*2*11 is in the sequence since it has 4 prime factors
72 = 2*2*2*3*3 is not in the sequence since it has 5 prime factors

Links

Crossrefs

Cf. A046304, A001222
For numbers with at most n prime factors: n=1: A000040, n=2: A037143, n=3: A037144, n=5: A166719

Programs

  • Mathematica
    Select[Range[100],PrimeOmega[#]<= 4 &] (* G. C. Greubel, May 24 2016 *)
  • PARI
    isA166718(n) = (bigomega(n) <= 4)

Formula

UNION of A000040, A001358, A014612, and A014613. - R. J. Mathar, Aug 08 2012

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

A345744 Numbers k such that k and k+1 are products of at least 5 primes.

Original entry on oeis.org

728, 944, 1215, 1376, 1539, 1700, 2024, 2079, 2295, 2511, 2624, 2672, 3087, 3104, 3159, 3320, 3375, 3807, 3824, 3968, 4095, 4374, 4940, 5103, 5264, 5480, 5535, 5624, 5750, 5775, 5967, 5984, 6075, 6344, 6399, 6560, 6831, 6875, 6975, 6992, 7208, 7424, 7695, 7749, 7856
Offset: 1

Views

Author

Tanya Khovanova, Jun 25 2021

Keywords

Comments

Integers k such that k and k+1 are in A046304. - Michel Marcus, Jun 26 2021

Examples

			728 = 2^3*7*13 is a product of 5 primes, while 729 = 3^6 is a product of 6 primes. Thus, 728 is in this sequence.

Crossrefs

Cf. A001222, A046304.

Programs

  • Maple
    q:= n-> andmap(x-> numtheory[bigomega](x)>4, [n, n+1]):
select(q, [$1..8000])[];  # Alois P. Heinz, Jun 26 2021
  • Mathematica
    Select[Range[10000], Total[Transpose[FactorInteger[#]][[2]]] > 4 && Total[Transpose[FactorInteger[# + 1]][[2]]] > 4 &]
  • PARI
    isok(k) = (bigomega(k) >= 5) && (bigomega(k+1) >= 5); \\ Michel Marcus, Jun 26 2021
  • Python
    from sympy import factorint
def ok(n): return all(sum(factorint(n+k).values()) > 4 for k in [0, 1])
print(list(filter(ok, range(8000)))) # Michael S. Branicky, Jun 25 2021
Showing 1-6 of 6 results.

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