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 1-9 of 9 results.

A075036 Smaller of two smallest consecutive numbers with 2n divisors.

Original entry on oeis.org

2, 14, 44, 104, 2511, 735, 29888, 2295, 6075, 5264, 2200933376, 5984, 689278976, 156735, 180224, 21735, 2035980763136, 223244, 9399153082499072, 458864, 41680575, 701443071, 2503092614937444351, 201824, 2707370000, 29785673727, 46977524, 5475519, 1737797404898095794225152
Offset: 1

Views

Author

Amarnath Murthy, Sep 03 2002

Keywords

Comments

There cannot be two consecutive numbers with the same odd number of divisors as both cannot be squares.
These numbers have the property that a(n) * (a(n) + 1) has 4*n^2 divisors. - David A. Corneth, Jun 24 2016
Conjecture: if a term k is even, the highest p-adic order of k (the maximum may be attained by several p's) occurs at p=2 and the highest p-adic order of k+1 occurs at p=3. If a term k is odd, the highest p-adic order of k occurs at p=3 and the highest p-adic order of k+1 occurs at p=2. - Chai Wah Wu, Mar 12 2019
a(49) = 378401464109375, a(58) = 79921490583489592950783. - Jon E. Schoenfield, May 07 2022
a(51) = 34210814718574592, a(55) = 2481402804069375, a(57) = 394311388855795712. - Jon E. Schoenfield, Nov 06 2023 - Nov 08 2023

Examples

			a(4) = 104 as tau(104) = tau(105) = 8.

Links

Crossrefs

Cf. A000005, A005237, A039832, A049103, A174456, A215197, A215199, A274357, A274358, A274359.

Programs

  • Mathematica
    a[n_] := (For[k=1, ! (DivisorSigma[0, k] == 2*n && DivisorSigma[0, k+1] == 2*n), k++]; k); Array[a, 10] (* Giovanni Resta, Jun 24 2016 *)
  • PARI
    a(n) = my(k=1); while(numdiv(k)!=2*n || numdiv(k+1)!=2*n, k++); k \\ Felix Fröhlich, Jun 24 2016

Formula

a(n) <= A215199(n-1) for n > 1. Conjecture: if p is prime, then a(p) = A215199(p-1). This conjecture is true if the conjecture in A215199 is true. The b-file of A215199 thus shows that a(p) = A215199(p-1) for prime p < 1279. - Chai Wah Wu, Mar 12 2019

Extensions

a(5)-a(24) from Max Alekseyev, Mar 12 2009
a(25)-a(28) from Giovanni Resta, Jun 24 2016
a(29) from Chai Wah Wu, Mar 12 2019

A356744 Numbers k such that both k and k+2 have exactly 8 divisors.

Original entry on oeis.org

40, 54, 102, 128, 136, 152, 182, 184, 230, 246, 248, 374, 424, 470, 472, 534, 582, 663, 710, 806, 822, 824, 854, 872, 902, 904, 999, 1105, 1192, 1256, 1309, 1334, 1336, 1432, 1446, 1526, 1542, 1545, 1576, 1593, 1645, 1686, 1784, 1832, 1864, 1885, 1910, 1928, 2006, 2013
Offset: 1

Views

Author

Jianing Song, Aug 25 2022

Keywords

Examples

			54 is a term since 54 and 56 both have 8 divisors.

Links

Crossrefs

Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), A356744 (m=6), this sequence (m=8).
Cf. also A274357 (numbers k such that k and k+1 both have exactly 8 divisors).

Programs

  • PARI
    isA356744(n) = numdiv(n)==8 && numdiv(n+2)==8

A274358 Numbers n such that n and n+1 both have 14 divisors.

Original entry on oeis.org

29888, 109375, 436671, 716607, 2829248, 3329343, 3948992, 5195583, 5568831, 8801216, 9767871, 10667456, 10947392, 12347072, 12627008, 14713407, 14959808, 16359488, 17479232, 20032191, 20278592, 20558528, 20965311, 23077952, 23544512, 24109375, 24477632
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Links

Crossrefs

Intersection of A005237 and A030632.
Cf. A000005, A274357.

Programs

  • PARI
    is(n)=numdiv(n)==14 && numdiv(n+1)==14
  • PARI
    list(lim)=my(v=List(),p6,t); forprime(p=2,sqrtnint(lim\2,6), p6=p^6; forprime(q=2,lim\p6, if(p==q,next); t=p6*q; if(numdiv(t+1)==14, listput(v,t)); if(numdiv(t-1)==14, listput(v,t-1)))); Set(v)

A274359 Numbers n such that n and n+1 both have 16 divisors.

Original entry on oeis.org

2295, 3255, 4185, 5480, 5642, 5655, 6104, 6279, 6344, 6669, 6783, 7160, 7314, 7749, 8294, 8295, 8645, 9176, 9344, 9624, 10184, 10664, 10695, 10856, 10934, 10989, 11319, 11528, 11570, 12440, 13034, 13064, 13208, 13310, 13629, 13695, 13845, 13959
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Links

Crossrefs

Intersection of A005237 and A030634.
Cf. A000005, A274357.

Programs

  • PARI
    is(n)=numdiv(n)==16 && numdiv(n+1)==16

A274362 Numbers n such that n and n+1 both have 24 divisors.

Original entry on oeis.org

5984, 11780, 20349, 22815, 24794, 26144, 27675, 29799, 31724, 33579, 33824, 34335, 34748, 36764, 37323, 37664, 38324, 38367, 38675, 38709, 40544, 41624, 42020, 44505, 44954, 47564, 47684, 48950, 50024, 51204, 52155, 52767, 53703, 53955, 54495, 55419
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Comments

Goldston-Graham-Pintz-Yildirim prove that this sequence is infinite; in particular infinitely often a(k) = A189982(n) = A189982(n+1) - 1. In fact, their proof shows that at least one of the residue classes 355740n + 47480, 889350n + 118700, or 592900n + 79134 contains infinitely many terms of this sequence.

Links

Crossrefs

Intersection of A005237 and A137487.
Cf. A000005, A274357, A189982.

Programs

  • Mathematica
    Reap[For[k = 1, k < 56000, k++, If[DivisorSigma[0, k] == DivisorSigma[0, k + 1] == 24, Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 16 2018 *)
  • PARI
    is(n)=numdiv(n)==24 && numdiv(n+1)==24

A274360 Numbers n such that n and n+1 both have 18 divisors.

Original entry on oeis.org

6075, 23275, 25856, 26900, 33524, 45324, 46475, 61299, 61347, 77076, 82075, 93924, 96236, 107775, 111924, 117324, 118700, 133524, 137924, 155771, 209524, 210176, 219275, 229275, 230643, 234099, 257724, 258475, 272924, 275300, 278271, 312987, 325899, 332667, 348524
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Links

Crossrefs

Intersection of A005237 and A030636.
Cf. A000005, A274357.

Programs

  • PARI
    is(n)=numdiv(n)==18 && numdiv(n+1)==18
  • PARI
    list(lim)=my(v=List()); forprime(p=2,sqrtint(lim\12), forprime(q=2,sqrtint(lim\p^2\2), if(p==q,next); my(pq2=(p*q)^2); forprime(r=2,lim\pq2, if(p==r || q==r, next); t=pq2*r; if(numdiv(t-1)==18, listput(v,t-1)); if(numdiv(t+1)==18, listput(v,t))))); Set(v)

A274361 Numbers n such that n and n+1 both have 20 divisors.

Original entry on oeis.org

5264, 7695, 13040, 27135, 33615, 38960, 41391, 44144, 54351, 55808, 62127, 64719, 70064, 72495, 77679, 100624, 101007, 108783, 108944, 116720, 124335, 124496, 132111, 132272, 139887, 145232, 160784, 165807, 176336, 186704, 191888, 199375, 202095
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Links

Crossrefs

Intersection of A005237 and A030638.
Cf. A000005, A274357.

Programs

  • PARI
    is(n)=numdiv(n)==20 && numdiv(n+1)==20

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)

A274365 Numbers n such that n and n+1 both have 30 divisors.

Original entry on oeis.org

180224, 257499, 579375, 1075599, 1990575, 2353616, 5598800, 10320624, 11560400, 13975983, 16951599, 17213552, 17651600, 17672499, 17784207, 20626991, 20660624, 21041775, 21912848, 22252400, 24533199, 24953103, 26161875, 26604207, 29232175, 29253392
Offset: 1

Views

Author

Charles R Greathouse IV, Jun 19 2016

Keywords

Links

Crossrefs

Intersection of A005237 and A137493.
Cf. A000005, A274357.

Programs

  • Mathematica
    SequencePosition[Table[If[DivisorSigma[0,n]==30,1,0],{n,3*10^7}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 10 2018 *)
  • PARI
    is(n)=numdiv(n)==30 && numdiv(n+1)==30
  • PARI
    list(lim)=my(v=List(),t); forprime(p=2,sqrtnint(lim\2,14), my(p14=p^14); forprime(q=2,lim\p14, if(p==q, next); t=p14*q; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1)))); forprime(p=2,sqrtnint(lim\4,9), my(p9=p^9); forprime(q=2,sqrtint(lim\p9), if(p==q, next); t=p9*q^2; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1)))); forprime(p=2,sqrtnint(lim\16,5), my(p5=p^5); forprime(q=2,sqrtnint(lim\p5,4), if(p==q, next); t=p5*q^4; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1)))); forprime(p=2,sqrtnint(lim\12,4), my(p4=p^4); forprime(q=2,sqrtint(lim\p4\2), if(p==q, next); my(q2=q^2); forprime(r=2,lim\p4\q2, if(p==r || q==r, next); t=p4*q2*r; if(numdiv(t+1)==30, listput(v,t)); if(numdiv(t-1)==30, listput(v,t-1))))); Set(v)
Showing 1-9 of 9 results.

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