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.

Previous Showing 11-14 of 14 results.

A334471 a(n) = Product_{d|n} (A069934(n) / sigma(d)) where A069934(n) = lcm_{d|n} sigma(d).

Original entry on oeis.org

1, 3, 4, 441, 6, 144, 8, 385875, 2704, 324, 12, 12446784, 14, 576, 576, 37418184916875, 18, 197413632, 20, 42007896, 1024, 1296, 24, 38118276000000, 34596, 1764, 35152000, 99574272, 30, 26873856, 32, 1409355934894096875, 2304, 2916, 2304, 1695648500686393344
Offset: 1

Views

Author

Jaroslav Krizek, May 01 2020

Keywords

Examples

			For n = 6; divisors d of 6: {1, 2, 3, 6}; sigma(d): {1, 3, 4, 12}; lcm_{d|6} sigma(d) = 12; a(6) = 12/1 * 12/3 * 12/4 * 12/12 = 144.

Crossrefs

Cf. Similar sequence with tau(d): A334470.
Cf. A000005, A000040, A069934, A206032.

Programs

  • Magma
    [&*[ LCM([&+Divisors(d): d in Divisors(n)]) / &+Divisors(d): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := (LCM @@ (s = DivisorSigma[1, Divisors[n]]))^Length[s] / Times @@ s; Array[a, 36] (* Amiram Eldar, May 02 2020 *)
  • PARI
    a(n) = {my(d=divisors(n), lcms = lcm(vector(#d, k, sigma(d[k])))); vecprod(vector(#d, k, lcms/sigma(d[k])));} \\ Michel Marcus, May 02 2020

Formula

a(n) = ((lcm_{d|n} sigma(d))^tau(n)) / Product_{d|n} (sigma(d)).
a(n) = A069934(n)^A000005(n) / A206032(n).
a(p) = p + 1 for p = primes (A000040).

A324987 a(n) = Product_{d|n} (tau(d)*sigma(d)) where tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 6, 8, 126, 12, 2304, 16, 7560, 312, 5184, 24, 8128512, 28, 9216, 9216, 1171800, 36, 21026304, 40, 27433728, 16384, 20736, 48, 234101145600, 1116, 28224, 49920, 65028096, 60, 110075314176, 64, 442940400, 36864, 46656, 36864, 60754075619328, 76, 57600, 50176
Offset: 1

Views

Author

Jaroslav Krizek, Mar 23 2019

Keywords

Comments

n divides a(n) for n: 1, 2, 6, 8, 12, 18, 24, 28, 36, 40, 48, 54, 56, 72, 80, 84, 96, 108, 112, 117, ...

Examples

			a(6) = (tau(1)*sigma(1)) * (tau(2)*sigma(2)) * (tau(3)*sigma(3)) * (tau(6)*sigma(6)) = (1*1) * (2*3) * (2*4) * (4*12) = 2304.

Crossrefs

Cf. A000005, A000203, A064840, A206032, A211776, A324986.

Programs

  • Magma
    [&*[NumberOfDivisors(d) * SumOfDivisors(d): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    Table[Product[DivisorSigma[0, k]*DivisorSigma[1, k], {k, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, Mar 23 2019 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, numdiv(d[k])*sigma(d[k])); \\ Michel Marcus, Mar 23 2019

Formula

a(n) = Product_{d|n} tau(d) * Product_{d|n} sigma(d) = A211776(n) * A206032(n).
a(p) = 2*(p + 1) for p = primes (A000040).
a(n) = Product_{d|n} A064840(d). - Antti Karttunen, Mar 28 2019

A280087 Numbers n such that Product_{d|n} sigma(d) = Product_{d|n+1} sigma(d).

Original entry on oeis.org

14, 1334, 1634, 2685, 33998, 42818, 84134, 122073, 166934, 289454, 383594, 440013, 544334, 605985, 649154, 655005, 1642154, 2284814, 2913105, 3571905, 3682622, 5181045, 6771405, 10074477, 10195305, 12825266, 15751533, 17714486, 17727554, 19886385, 25096665, 33422277, 34577834, 34883654
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2016

Keywords

Comments

sigma(n) is the sum of the divisors of n (A000203).
Numbers n such that A206032(n) = A206032(n+1).

Examples

			14 is a term because Product_{d|14} sigma(d) = 1 * 3 * 8 * 24 = Product_{d|15} sigma(d) = 1 * 4 * 6 * 24 = 576.

Crossrefs

Cf. A000203, A206032.

Programs

  • Magma
    [n: n in [1..1000] | &*[SumOfDivisors(d): d in Divisors(n)]  eq &*[SumOfDivisors(d): d in Divisors(n+1)]]
  • Mathematica
    Select[Range[5000], Times @@ DivisorSigma[1, Divisors[#]] == Times @@ DivisorSigma[1, Divisors[# + 1]] &] (* G. C. Greubel, Dec 26 2016 *)
  • PARI
    isok(n) = my(d = divisors(n), dd = divisors(n+1)); prod(k=1, #d, sigma(d[k])) == prod(k=1, #dd, sigma(dd[k])); \\ Michel Marcus, Dec 26 2016

Extensions

More terms from Michel Marcus, Dec 26 2016

A324980 a(n) = Product_{d|n} (d*sigma(d)) where sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 6, 12, 168, 30, 5184, 56, 20160, 1404, 32400, 132, 48771072, 182, 112896, 129600, 9999360, 306, 425782656, 380, 762048000, 451584, 627264, 552, 8427641241600, 23250, 1192464, 1516320, 4956585984, 870, 21767823360000, 992, 20158709760, 2509056, 3370896
Offset: 1

Views

Author

Jaroslav Krizek, Mar 22 2019

Keywords

Examples

			a(6) = 1*sigma(1) * 2*sigma(2) * 3*sigma(3) * 6*sigma(6) = (1*1) * (2*3) * (3*4) * (6*12) = 5184.

Crossrefs

Cf. A000203, A001001 (Sum_{d|n} (d*sigma(d))), A206032.

Programs

  • Magma
    [&*[d * SumOfDivisors(d): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    Array[Times @@ Map[# DivisorSigma[1, #] &, Divisors@ #] &, 34] (* Michael De Vlieger, Mar 24 2019 *)
  • PARI
    a(n) = my(p=1); fordiv(n, d, p *= d*sigma(d)); p; \\ Michel Marcus, Mar 22 2019

Formula

a(n) = (Product_{d|n} d) * (Product_{d|n} sigma(d)) = A007955(n) * A206032(n).
a(p) = p*(p+1) for p = primes (A000040).
Previous Showing 11-14 of 14 results.

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

Content is available under The OEIS End-User License Agreement.