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 1-4 of 4 results.

A083538 a(n) = sigma(n)*sigma(n+1)/gcd(sigma(n+1), sigma(n))^2.

Original entry on oeis.org

3, 12, 28, 42, 2, 6, 120, 195, 234, 6, 21, 2, 84, 1, 744, 558, 78, 780, 210, 336, 72, 6, 10, 1860, 1302, 420, 35, 420, 60, 36, 2016, 336, 72, 72, 4368, 3458, 570, 210, 1260, 105, 112, 264, 231, 182, 156, 6, 372, 7068, 589, 744, 1764, 1323, 180, 15, 15, 6, 72, 6, 70
Offset: 1

Views

Author

Labos Elemer, May 21 2003

Keywords

Comments

a(n) = A060781(n)/A060780(n) = A083539(n)/A060780(n)^2; quotient when lcm(sigma(n+1), sigma(n)) is divided by gcd(sigma(n+1), sigma(n)).

Examples

			n=10: sigma(10)=18, sigma(11)=12, lcm(18, 12)=36, gcd(18, 12)=6, a(10) = 36/6 = 6.

Links

Crossrefs

Cf. A000010, A066813, A058515, A000203, A060781, A060780, A083539.

Programs

  • Mathematica
    f[x_] := DivisorSigma[1, x] t=Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 1, 128}]
Times@@#/(GCD@@#)^2&/@Partition[DivisorSigma[1,Range[60]],2,1] (* Harvey P. Dale, Feb 17 2016 *)
  • PARI
    a(n)=my(x=sigma(n),y=sigma(n+1)); x*y/gcd(x,y)^2 \\ Charles R Greathouse IV, Mar 09 2014

Extensions

Edited by N. J. A. Sloane, Apr 29 2007
Corrections by Charles R Greathouse IV, Mar 09 2014

A083539 a(n) = sigma(n) * sigma(n+1): product of sigma-values for consecutive integers.

Original entry on oeis.org

3, 12, 28, 42, 72, 96, 120, 195, 234, 216, 336, 392, 336, 576, 744, 558, 702, 780, 840, 1344, 1152, 864, 1440, 1860, 1302, 1680, 2240, 1680, 2160, 2304, 2016, 3024, 2592, 2592, 4368, 3458, 2280, 3360, 5040, 3780, 4032, 4224, 3696, 6552, 5616, 3456, 5952
Offset: 1

Views

Author

Labos Elemer, May 21 2003

Keywords

Links

Crossrefs

Cf. A000203, A002378, A060781, A060780, A083538, A330322 (partial sums).
Cf. A083481, A083542, A092517.

Programs

  • Mathematica
    f[x_] := DivisorSigma[1, x]; t=Table[f[w+1]*f[w], {w, 1, 128}]
Times@@@Partition[DivisorSigma[1,Range[50]],2,1] (* Harvey P. Dale, May 21 2014 *)
  • PARI
    a(n)=sigma(n)*sigma(n+1) \\ Charles R Greathouse IV, Feb 14 2013

Formula

a(n) = A000203(A002378(n)). - Amiram Eldar, Jul 10 2024

A083540 Numbers k such that A083539(k) is a square; solutions x to sigma(x+1)*sigma(x)=y^2 for some y.

Original entry on oeis.org

14, 30, 51, 161, 186, 206, 223, 329, 552, 713, 759, 869, 957, 994, 995, 1248, 1334, 1364, 1634, 1715, 1819, 2093, 2133, 2584, 2685, 2820, 2821, 2974, 3115, 3145, 3485, 4212, 4308, 4312, 4364, 4408, 4649
Offset: 1

Views

Author

Labos Elemer, May 21 2003

Keywords

Examples

			x=30: sigma(30)=72, sigma(31)=32, product = 72*32 = 256*9 = 24^2.

Links

Crossrefs

Cf. A000203, A083539, A060780, A060781, A083538.

Programs

  • Mathematica
    Do[s=Sqrt[DivisorSigma[1, n+1]*DivisorSigma[1, n]]; If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Flatten[Position[Times@@@Partition[DivisorSigma[1,Range[5000]],2,1], ?(IntegerQ[Sqrt[#]]&)]] (* _Harvey P. Dale, Mar 07 2016 *)

A083541 Values of y from solutions to sigma(x+1)*sigma(x)=y^2, where A083539(x) = y^2 is a square number.

Original entry on oeis.org

24, 48, 84, 264, 288, 312, 336, 576, 960, 1152, 1440, 1440, 1440, 1440, 1680, 2100, 2160, 2688, 2640, 3360, 3024, 3360, 3360, 4320, 4320, 5376, 4032, 4464, 5040, 4788, 6048, 7392, 6720, 6840, 7644, 6300, 7440, 7560, 7020, 10080, 10080, 8064, 10080
Offset: 1

Views

Author

Labos Elemer, May 21 2003

Keywords

Links

Crossrefs

Cf. A000203, A083539, A083540, A083538, A060780, A060781.

Programs

  • Mathematica
    Do[s=Sqrt[DivisorSigma[1, n+1]*DivisorSigma[1, n]]; If[IntegerQ[s], Print[s]], {n, 1, 5000}]

Formula

a(n) = sqrt(sigma(A083540(n)) * sigma(1+A083540(n))).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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