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.

A175747 Numbers with 38 divisors.

Original entry on oeis.org

786432, 1310720, 1835008, 2883584, 3407872, 4456448, 4980736, 6029312, 7602176, 8126464, 9699328, 10747904, 11272192, 12320768, 13893632, 15466496, 15990784, 17563648, 18612224, 19136512, 20709376, 21757952, 23330816, 25427968, 26476544, 27000832, 28049408
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the forms p^37 and p^18*q^1, where p and q are distinct primes.

Links

Crossrefs

Cf. A175745, A175746, A139572.

Programs

  • Mathematica
    Select[Range[10000000],DivisorSigma[0,#]==38&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)
  • PARI
    is(n)=numdiv(n)==38 \\ Charles R Greathouse IV, Jun 19 2016
  • Python
    def A175747(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 int(n+x-sum(primepi(x//p**18) for p in primerange(integer_nthroot(x,18)[0]+1))+primepi(integer_nthroot(x,19)[0])-primepi(integer_nthroot(x,37)[0]))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

Formula

A000005(a(n))=38.

Extensions

Extended by T. D. Noe, May 08 2011

A190464 Numbers with prime factorization p^4*q^6.

Original entry on oeis.org

5184, 11664, 40000, 153664, 250000, 455625, 937024, 1265625, 1750329, 1827904, 1882384, 5345344, 8340544, 9529569, 10673289, 17909824, 20820969, 28344976, 37515625, 45265984, 59105344, 60886809, 73530625, 77228944, 95004009, 119946304, 143496441, 180848704, 204004089, 218803264
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Comments

A subsequence of A175745 (Numbers with 35 divisors).
First different term in A175745 is 17179869184(=2^34).

Links

Crossrefs

Cf. A030629, A179692, A179699, A179705, A175745.
Cf. A085964, A085966.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={4,6}; Select[Range[50000000],f] (*and*) lst={};Do[Do[If[n!=m,AppendTo[lst,Prime[n]^6*Prime[m]^4]], {n,50}],{m,50}]; Take[Union@lst,50]
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/6), t=p^6;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Formula

Sum_{n>=1} 1/a(n) = P(4)*P(6) - P(10) = A085964 * A085966 - P(10) = 0.000320..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A336595 Numbers whose number of divisors is divisible by 5.

Original entry on oeis.org

16, 48, 80, 81, 112, 144, 162, 176, 208, 240, 272, 304, 324, 336, 368, 400, 405, 432, 464, 496, 512, 528, 560, 567, 592, 624, 625, 648, 656, 688, 720, 752, 784, 810, 816, 848, 880, 891, 912, 944, 976, 1008, 1040, 1053, 1072, 1104, 1134, 1136, 1168, 1200, 1232
Offset: 1

Views

Author

Amiram Eldar, Jul 26 2020

Keywords

Comments

The asymptotic density of this sequence is 1 - zeta(5)/zeta(4) = 0.0419426259... (Sathe, 1945).

Examples

			16 is a term since A000005(16) = 5 is divisible by 5.

References

  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 63.

Links

Crossrefs

Cf. A000005, A008587, A059269, A336596.
Cf. A030514, A030628, A113849, A178739, A179665, A030633, A030638, A137488, A137493, A175745.

Programs

  • Maple
    q:= n-> is(irem(numtheory[tau](n), 5)=0):
select(q, [$1..1300])[];  # Alois P. Heinz, Jul 26 2020
  • Mathematica
    Select[Range[1300], Divisible[DivisorSigma[0, #], 5] &]

Formula

A030514 UNION A030628 \ {1} UNION A030633 UNION A030638 UNION A137488 UNION A137493 UNION A175745 UNION A175749 UNION A175752 UNION A175756 UNION ... - R. J. Mathar, May 05 2023

A175746 Numbers with 36 divisors.

Original entry on oeis.org

1260, 1440, 1800, 1980, 2016, 2100, 2340, 2400, 2700, 2772, 2940, 3060, 3150, 3168, 3276, 3300, 3420, 3528, 3744, 3840, 3900, 4140, 4284, 4410, 4500, 4704, 4788, 4860, 4896, 4950, 5100, 5148, 5220, 5292, 5376, 5472, 5580, 5600, 5700, 5796, 5850, 6468, 6624
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the forms p^35, p^17*q^1, p^11*q^2, p^8*q^3, p^5*q^5, p^8*q^1*r^1, p^5*q^2*r^1, p^3*q^2*r^2 and p^2*q^2*r^1*s^1, where p, q, r, and s are distinct primes.

Links

Crossrefs

Cf. A175744, A175745.

Programs

Formula

A000005(a(n))=36.

Extensions

Extended by T. D. Noe, May 08 2011

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 1-5 of 5 results.

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