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.

A336723 a(n) = lcm(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 6, 12, 168, 30, 36, 56, 960, 351, 900, 132, 12096, 182, 1176, 1800, 158720, 306, 75816, 380, 168000, 14112, 4356, 552, 1658880, 11625, 14196, 29160, 65856, 870, 810000, 992, 2064384, 17424, 31212, 58800, 917070336, 1406, 21660, 85176, 23040000, 1722, 6223392
Offset: 1

Views

Author

Jaroslav Krizek, Aug 01 2020

Keywords

Comments

a(n) = pod(n) for numbers n: 1, 6, 30, 66, 84, 102, 120, 210, 270, 318, 330, 420, 462, 510, 546, 570, 642, ...

Examples

			a(6) = lcm(tau(6), sigma(6), pod(6)) = lcm(4, 12, 36) = 36.

Crossrefs

Cf. A009278 (lcm(tau(n), sigma(n))), A324528 (lcm(tau(n), pod(n))), A324529 (lcm(sigma(n), pod(n))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336722 (gcd(tau(n), sigma(n), pod(n))).
Cf. A277521 (numbers k such that a(k) = pod(k) and simultaneously A336722(k) = tau(k)).

Programs

  • Magma
    [LCM([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];
  • Mathematica
    a[n_] := LCM @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 50] (* Amiram Eldar, Aug 01 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); lcm([#d, vecsum(d), vecprod(d)]); \\ Michel Marcus, Aug 12 2020

Formula

a(p) = p^2 + p for p = primes (A000040).

A334793 a(n) = Sum_{d|n} lcm(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 3, 7, 27, 11, 45, 15, 91, 34, 113, 23, 1797, 27, 213, 917, 5211, 35, 5904, 39, 24137, 1785, 509, 47, 333637, 386, 705, 2950, 66093, 59, 811055, 63, 103515, 4385, 1193, 4925, 10085352, 75, 1485, 6117, 2584201, 83, 3113715, 87, 256085, 183194, 2165, 95
Offset: 1

Views

Author

Jaroslav Krizek, May 12 2020

Keywords

Examples

			a(6) = lcm(tau(1), pod(1)) + lcm(tau(2), pod(2)) + lcm(tau(3), pod(3)) + lcm(tau(6), pod(6)) = lcm(1, 1) + lcm(2, 2) + lcm(2, 3) + lcm(4, 36) = 1 + 2 + 6 + 36 = 45.

Links

Crossrefs

Cf. A334662 (Sum_{d|n} gcd(tau(d), pod(d))), A334784 (Sum_{d|n} lcm(tau(d), sigma(d))).
Cf. A000005 (tau(n)), A007955 (pod(n)), A324528 (lcm(tau(n), pod(n))).

Programs

  • Magma
    [&+[LCM(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]];
  • Maple
    pod:= proc(n) option remember; convert(numtheory:-divisors(n),`*`) end proc:
f:= proc(n) local d; add(ilcm(numtheory:-tau(d), pod(d)),d=numtheory:-divisors(n)) end proc:
map(f, [$1..100]); # Robert Israel, Jan 02 2025
  • Mathematica
    a[n_] := DivisorSum[n, LCM[(d = DivisorSigma[0, #]), #^(d/2)] &]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)
  • PARI
    a(n) = sumdiv(n, d, lcm(numdiv(d), vecprod(divisors(d)))); \\ Michel Marcus, May 12 2020

Formula

a(p) = 2p + 1 for p = odd primes (A065091).

A307892 a(n) = lcm(tau(n), pod(n)) / n, where tau(k) = the number of divisors of k (A000005) and pod(n) = the product of divisors of k (A007955).

Original entry on oeis.org

1, 1, 2, 6, 2, 6, 2, 8, 3, 10, 2, 144, 2, 14, 60, 320, 2, 324, 2, 1200, 84, 22, 2, 13824, 15, 26, 108, 2352, 2, 27000, 2, 3072, 132, 34, 140, 279936, 2, 38, 156, 64000, 2, 74088, 2, 5808, 4050, 46, 2, 26542080, 21, 7500, 204, 8112, 2, 157464, 220, 175616, 228
Offset: 1

Views

Author

Jaroslav Krizek, May 03 2019

Keywords

Comments

n divides lcm(tau(n), pod(n)) for all n >= 1.

Examples

			For n=4: a(4) = lcm(tau(4), pod(4))/4 = lcm(3, 8)/4 = 24/4 = 6.

Links

Crossrefs

Cf. A000005, A007955, A324528.

Programs

  • Magma
    [LCM(NumberOfDivisors(n), &*[d: d in Divisors(n)]) / n: n in [1.. 10^5]];
  • Mathematica
    Table[(LCM[DivisorSigma[0,n],Times@@Divisors[n]])/n,{n,60}] (* Harvey P. Dale, Sep 22 2024 *)

Formula

a(n) = A324528(n) / n.

A334807 a(n) = Product_{d|n} lcm(tau(d), pod(d)) where tau(k) is the number of divisors of k (A000005) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 2, 6, 48, 10, 432, 14, 3072, 162, 2000, 22, 17915904, 26, 5488, 54000, 15728640, 34, 68024448, 38, 1152000000, 148176, 21296, 46, 380420285792256, 3750, 35152, 472392, 8674025472, 58, 314928000000000, 62, 1546188226560, 574992, 78608, 686000
Offset: 1

Views

Author

Jaroslav Krizek, Jun 26 2020

Keywords

Examples

			a(6) = lcm(tau(1), pod(1)) * lcm(tau(2), pod(2)) * lcm(tau(3), pod(3)) * lcm(tau(6), pod(6)) = lcm(1, 1) * lcm(2, 2) * lcm(2, 3) * lcm(4, 36) = 1 * 2 * 6 * 36 = 432.

Links

Crossrefs

Cf. A334793 (Sum_{d|n} lcm(tau(d), pod(d))), A334730 (Product_{d|n} gcd(tau(d), pod(d))).
Cf. A000005 (tau(n)), A007955 (pod(n)), A324528 (lcm(tau(n), pod(n))).

Programs

  • Magma
    [&*[LCM(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]];
  • Maple
    pod:= proc(n) option remember; convert(numtheory:-divisors(n),`*`) end proc:
f:= proc(n) local d; mul(ilcm(numtheory:-tau(d), pod(d)),d=numtheory:-divisors(n)) end proc:
map(f, [$1..50]); # Robert Israel, Jan 02 2025
  • Mathematica
    a[n_] := Product[LCM[DivisorSigma[0, d], Times @@ Divisors[d]], {d, Divisors[n]}]; Array[a, 35] (* Amiram Eldar, Jun 27 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(numdiv(d[k]), vecprod(divisors(d[k])))); \\ Michel Marcus, Jun 27 2020

Formula

a(p) = 2p for p = odd primes (A065091).

A338563 a(n) = lcm(n, tau(n), sigma(n)) where tau(k) is the number of divisors of k (A000005) and sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 6, 12, 84, 30, 12, 56, 120, 117, 180, 132, 84, 182, 168, 120, 2480, 306, 234, 380, 420, 672, 396, 552, 120, 2325, 1092, 1080, 168, 870, 360, 992, 2016, 528, 1836, 1680, 3276, 1406, 1140, 2184, 360, 1722, 672, 1892, 924, 1170, 1656, 2256, 7440, 2793, 4650
Offset: 1

Views

Author

Jaroslav Krizek, Nov 02 2020

Keywords

Examples

			a(6) = lcm(6, tau(6), sigma(6)) = lcm(6, 4, 12) = 12.

Crossrefs

Cf. A000005, A000203.
Cf. A337323 (gcd(n, tau(n), sigma(n))), A324528 (lcm(n, tau(n), pod(n))), A324529 (lcm(n, sigma(n), pod(n))).

Programs

  • Magma
    [LCM([n, #Divisors(n), &+Divisors(n)]): n in [1..100]]
  • Mathematica
    a[n_] := LCM @@ {n, DivisorSigma[0, n], DivisorSigma[1, n]}; Array[a, 50] (* Amiram Eldar, Nov 03 2020 *)
  • PARI
    a(n) = my(f=factor(n)); lcm([n, sigma(f), numdiv(f)]); \\ Michel Marcus, Nov 03 2020

Formula

a(p) = p *(p + 1) for p = primes (A000040).
Showing 1-5 of 5 results.

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