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.

A166546 Natural numbers n such that d(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 86, 87, 89, 90
Offset: 1

Views

Author

Giovanni Teofilatto, Oct 16 2009

Keywords

Comments

Natural numbers n such that d(d(n)+1)= 2. - Giovanni Teofilatto, Oct 26 2009
The complement is the union of A001248, A030514, A030516, A030626, A030627, A030629, A030631, A030632, A030633 etc. - R. J. Mathar, Oct 26 2009

Links

Crossrefs

Cf. A000005.
Cf. A073915. - R. J. Mathar, Oct 26 2009

Programs

  • Magma
    [n: n in [1..100] | IsPrime(NumberOfDivisors(n)+1)]; // Vincenzo Librandi, Jan 20 2019
  • Mathematica
    Select[Range@90, PrimeQ[DivisorSigma[0, #] + 1] &] (* Vincenzo Librandi, Jan 20 2019 *)
  • PARI
    isok(n) = isprime(numdiv(n)+1); \\ Michel Marcus, Jan 20 2019

Formula

{1} U A000040 U A030513 U A030515 U A030628 U A030630 U A030634 U A030636 U A137485 U A137491 U A137493 U ... . - R. J. Mathar, Oct 26 2009

A175742 Numbers with 32 divisors.

Original entry on oeis.org

840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2310, 2376, 2688, 2730, 2760, 2808, 2856, 2970, 3000, 3080, 3192, 3432, 3456, 3480, 3510, 3570, 3640, 3672, 3720, 3864, 3990, 4104, 4158, 4224, 4290, 4440, 4480, 4488, 4590, 4760, 4830, 4872
Offset: 1

Views

Author

Jaroslav Krizek, Aug 27 2010

Keywords

Comments

Numbers of the form p^31, p^15*q^1, p^7*q^3, p^7*q^1*r^1, p^3*q^3*r^1, p^3*q^1*r^1*s^1 and p^1*q^1*r^1*s^1*t^1, where p, q, r, s and t are distinct primes.

Links

Crossrefs

Cf. A137491, A137492, A137493, A139571.
Cf. A046303 (a subsequence). - Michel Marcus, Apr 06 2017

Programs

Formula

A000005(a(n))=32.

Extensions

Extended by T. D. Noe, May 09 2011

A036457 Numbers k for which exactly 5 applications of A000005 are needed to reach 2.

Original entry on oeis.org

60, 72, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 200, 204, 220, 224, 228, 234, 240, 252, 260, 276, 288, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 392, 396, 414, 416, 420, 432, 444, 450, 460, 468, 476, 480
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

Subsequences include A030630 (numbers with 12 divisors), A030636 (numbers with 18 divisors), A030638 (numbers with 20 divisors), A137491 (numbers with 28 divisors), etc. [edited by Jon E. Schoenfield, May 12 2018]

Examples

			a(13)=180; the successive iterates are 18, 6, 4, 3, and finally the 5th is 2;
a(3)=84; divisor numbers are 12, 6, 4, 3, and 2.

Links

Crossrefs

Cf. A000005, A007624, A036450, A036452, A036453, A036455, A030630.

Programs

  • Maple
    A036459:= proc(n) option remember;
  if n <= 2 then 0 else 1 + procname(numtheory:-tau(n)) fi
end proc:
select(A036459 = 5, [$1..1000]); # Robert Israel, Jan 25 2016
  • Mathematica
    Select[Range@ 480, Last@ # == 2 && #[[5]] != 2 &@ NestList[DivisorSigma[0, #] &, #, 5] &] (* Michael De Vlieger, Jan 26 2016 *)
  • PARI
    is(n)=for(i=1,4,n=numdiv(n); if(n<3, return(0))); numdiv(n)==2 \\ Charles R Greathouse IV, Sep 17 2015

Formula

d(d(d(d(d(a(n)))))) = 2 for all n.
A036459(a(n)) = 5. - Ivan Neretin, Jan 25 2016

Extensions

New name from Robert Israel, Jan 25 2016

A274364 Numbers n such that n and n+1 both have 28 divisors.

Original entry on oeis.org

156735, 809919, 1276479, 1429568, 1649727, 1836351, 2269376, 2489535, 2549312, 2769471, 2922560, 3202496, 3236031, 4449087, 4729023, 4915647, 5068736, 5288895, 5441984, 5628608, 5721920, 6035391, 6281792, 6468416, 6595263, 6841664, 7028288, 7248447, 7528383
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Links

Crossrefs

Intersection of A005237 and A137491.
Cf. A000005, A274357.

Programs

  • Mathematica
    SequencePosition[DivisorSigma[0,Range[7600000]],{28,28}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 10 2017 *)
  • PARI
    is(n)=numdiv(n)==28 && numdiv(n+1)==28
  • PARI
    list(lim)=my(v=List(),t,p6); forprime(p=2,sqrtnint(lim\6,6), p6=p^6; forprime(q=2,lim\p6\2, if(p==q,next); forprime(r=2,lim\p6\q, if(p==r || q==r, next); t=p6*q*r; if(numdiv(t+1)==28, listput(v,t)); if(numdiv(t-1)==28, listput(v,t-1))))); forprime(p=2,sqrtnint(lim\8,6), p6=p^6; forprime(q=2,sqrtnint(lim\p6,3), if(p==q, next); t=p6*q^3; if(numdiv(t+1)==28, listput(v,t));if(numdiv(t-1)==28, listput(v,t-1)))); Set(v)

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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