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 1-3 of 3 results.

A334782 a(n) = Sum_{d|n} lcm(d, tau(d)).

Original entry on oeis.org

1, 3, 7, 15, 11, 21, 15, 23, 16, 33, 23, 45, 27, 45, 77, 103, 35, 48, 39, 105, 105, 69, 47, 77, 86, 81, 124, 141, 59, 231, 63, 199, 161, 105, 165, 108, 75, 117, 189, 153, 83, 315, 87, 213, 176, 141, 95, 397, 162, 258, 245, 249, 107, 372, 253, 205, 273, 177
Offset: 1

Views

Author

Jaroslav Krizek, May 10 2020

Keywords

Examples

			a(6) = lcm(1, tau(1)) + lcm(2, tau(2)) + lcm(3, tau(3)) + lcm(6, tau(6)) = lcm(1, 1) + lcm(2, 2) + lcm(3, 2) + lcm(6, 4) = 1 + 2 + 6 + 12 = 21.

Crossrefs

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334783 (Sum_{d|n} lcm(d, sigma(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A009230 (lcm(n, tau(n))).

Programs

  • Magma
    [&+[LCM(d, #Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := DivisorSum[n, LCM[#, DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, May 10 2020 *)
  • PARI
    a(n) = sumdiv(n, d, lcm(d, numdiv(d))); \\ Michel Marcus, May 10 2020

Formula

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

A334784 a(n) = Sum_{d|n} lcm(tau(d), sigma(d)).

Original entry on oeis.org

1, 7, 5, 28, 7, 23, 9, 88, 44, 49, 13, 128, 15, 39, 35, 243, 19, 140, 21, 112, 45, 55, 25, 308, 100, 105, 84, 228, 31, 161, 33, 369, 65, 133, 63, 1064, 39, 87, 75, 532, 43, 183, 45, 160, 152, 103, 49, 1083, 66, 328, 95, 420, 55, 300, 91, 408, 105, 217, 61, 476
Offset: 1

Views

Author

Jaroslav Krizek, May 10 2020

Keywords

Examples

			a(6) = lcm(tau(1), sigma(1)) + lcm(tau(2), sigma(2)) + lcm(tau(3), sigma(3)) + lcm(tau(6), sigma(6)) = lcm(1, 1) + lcm(2, 3) + lcm(2, 4) + lcm(4, 12) = 1 + 6 + 4 + 12 = 23.

Crossrefs

Cf. A334579 (Sum_{d|n} gcd(tau(d), sigma(d))), A334783 (Sum_{d|n} lcm(d, sigma(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A009278 (lcm(tau(n), sigma(n))).

Programs

  • Magma
    [&+[LCM(#Divisors(d), &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := DivisorSum[n, LCM[DivisorSigma[0, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 10 2020 *)
  • PARI
    a(n) = sumdiv(n, d, lcm(numdiv(d), sigma(d))); \\ Michel Marcus, May 10 2020

Formula

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

A334805 a(n) = Product_{d|n} lcm(d, sigma(d)) where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 6, 12, 168, 30, 864, 56, 20160, 1404, 16200, 132, 2032128, 182, 56448, 43200, 9999360, 306, 23654592, 380, 190512000, 451584, 313632, 552, 29262643200, 23250, 596232, 1516320, 88510464, 870, 100776960000, 992, 20158709760, 836352, 1685448, 2822400
Offset: 1

Views

Author

Jaroslav Krizek, Jun 26 2020

Keywords

Examples

			a(6) = lcm(1, sigma(1)) * lcm(2, sigma(2)) * lcm(3, sigma(3)) * lcm(6, sigma(6)) = lcm(1, 1) * lcm(2, 3) * lcm(3, 4) * lcm(6, 12) = 1 * 6 * 12 * 12 = 864.

Crossrefs

Cf. A334783 (Sum_{d|n} lcm(d, sigma(d))), A334491 (Product_{d|n} gcd(d, sigma(d))).
Cf. A000203 (sigma(n)), A009242 (lcm(n, sigma(n))), A036690.

Programs

  • Magma
    [&*[LCM(d, &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := Product[LCM[d, DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 35] (* Amiram Eldar, Jun 27 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], sigma(d[k]))); \\ Michel Marcus, Jun 27 2020

Formula

a(p) = p^2 + p for p = primes (A000040).
Showing 1-3 of 3 results.

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