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.

A030638 Numbers with 20 divisors.

Original entry on oeis.org

240, 336, 432, 528, 560, 624, 648, 810, 816, 880, 912, 1040, 1104, 1134, 1232, 1360, 1392, 1456, 1488, 1520, 1536, 1776, 1782, 1840, 1904, 1968, 2000, 2064, 2106, 2128, 2256, 2288, 2320, 2480, 2544, 2560, 2576, 2754, 2832, 2835, 2928
Offset: 1

Views

Author

Jeff Burch

Keywords

Comments

Numbers of the form p^19, p*q^9 (A179692), p*q*r^4 (A179644) or p^3*q^4 (A179666), where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Links

Crossrefs

Cf. A030636, A030637, A137484.

Programs

A189990 Numbers with prime factorization p^2*q^6.

Original entry on oeis.org

576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, 62500, 87616, 88209, 107584, 118336, 123201, 140625, 141376, 179776, 210681, 222784, 238144, 263169, 287296, 322624, 341056, 385641, 399424, 440896, 470596
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 03 2011

Keywords

Links

Crossrefs

Subsequence of A137484.
Cf. A179645, A179664.
Cf. A065036, A085548, A085966, A085968.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={2,6}; Select[Range[800000],f]
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/6), t=p^6;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011
  • Python
    from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189990(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**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,8)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) = A085548 * A085966 - A085968 = 0.003658..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020
a(n) = A065036(n)^2. - Chai Wah Wu, Mar 27 2025

A175750 Numbers with 42 divisors.

Original entry on oeis.org

2880, 4032, 4800, 6336, 7488, 9408, 9792, 10944, 11200, 13248, 14580, 15552, 15680, 16704, 17600, 17856, 20412, 20800, 21312, 23232, 23328, 23616, 24768, 27072, 27200, 30400, 30528, 32076, 32448, 33984, 34496, 35136, 36450, 36800, 37908, 38592, 38720, 40768
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the forms p^41, p^20*q^1, p^13*q^2, p^6*q^5 and p^6*q^2*r^1 (A179703), where p, q, and r are distinct primes.

Links

Crossrefs

Cf. A137484, A137488.

Programs

Formula

A000005(a(n))=42.

Extensions

Extended by T. D. Noe, May 08 2011

A139588 Nonprime numbers with Fibonacci number of divisors.

Original entry on oeis.org

1, 4, 9, 16, 24, 25, 30, 40, 42, 49, 54, 56, 66, 70, 78, 81, 88, 102, 104, 105, 110, 114, 121, 128, 130, 135, 136, 138, 152, 154, 165, 169, 170, 174, 182, 184, 186, 189, 190, 195, 222, 230, 231, 232, 238, 246, 248, 250, 255, 258, 266, 273, 282, 285, 286, 289
Offset: 1

Views

Author

Omar E. Pol, May 09 2008

Keywords

Comments

A000005(a(n)) is a Fibonacci number.
The union of {1}, A001248, A030514, A030626, A030631, A137484, etc. [From R. J. Mathar, Oct 26 2009]

Links

Crossrefs

Cf. A000005, A000045, A018252, A123193, A139589, A139590.

Programs

  • Mathematica
    Module[{fibs=Fibonacci[Range[20]]},Select[Range[300],!PrimeQ[#]&&MemberQ[ fibs,DivisorSigma[0,#]]&]] (* Harvey P. Dale, Jan 20 2023 *)

Formula

A123193 \ A000040. [From R. J. Mathar, Oct 23 2009]

Extensions

More terms from R. J. Mathar, Oct 23 2009

A175756 Numbers with 50 divisors.

Original entry on oeis.org

6480, 9072, 14256, 16848, 22032, 24624, 29808, 30000, 37584, 40176, 41472, 47952, 53136, 55728, 60912, 68688, 70000, 76464, 79056, 86832, 92016, 94608, 101250, 102384, 107568, 110000, 115248, 115344, 125712, 130000, 130896, 133488
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the forms p^49, p^24*q^1, p^9*q^4 and p^4*q^4*r^1 (A190012), where p, q and r are distinct primes.

Links

Crossrefs

Cf. A137484, A137488, A175750.

Programs

Formula

A000005(a(n))=50.

A336596 Numbers whose number of divisors is divisible by 7.

Original entry on oeis.org

64, 192, 320, 448, 576, 704, 729, 832, 960, 1088, 1216, 1344, 1458, 1472, 1600, 1728, 1856, 1984, 2112, 2240, 2368, 2496, 2624, 2752, 2880, 2916, 3008, 3136, 3264, 3392, 3520, 3645, 3648, 3776, 3904, 4032, 4160, 4288, 4416, 4544, 4672, 4800, 4928, 5056, 5103
Offset: 1

Views

Author

Amiram Eldar, Jul 26 2020

Keywords

Comments

The asymptotic density of this sequence is 1 - zeta(7)/zeta(6) = 0.0088404638... (Sathe, 1945).

Examples

			64 is a term since A000005(64) = 7 is divisible by 7.

Links

Crossrefs

Cf. A000005, A008589, A059269, A336595.
Cf. A030516, A113851 and A138031 are subsequences.

Programs

  • Maple
    q:= n-> is(irem(numtheory[tau](n), 7)=0):
select(q, [$1..5500])[];  # Alois P. Heinz, Jul 26 2020
  • Mathematica
    Select[Range[5000], Divisible[DivisorSigma[0, #], 7] &]

Formula

A030516 UNION A030632 UNION A137484 UNION A137491 UNION A175745 UNION A175750 UNION ... - R. J. Mathar, May 05 2023
Showing 1-6 of 6 results.

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