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.

A334579 a(n) = Sum_{d|n} gcd(tau(d), sigma(d)).

Original entry on oeis.org

1, 2, 3, 3, 3, 8, 3, 4, 4, 6, 3, 11, 3, 8, 9, 5, 3, 12, 3, 13, 9, 8, 3, 16, 4, 6, 8, 11, 3, 24, 3, 8, 9, 6, 9, 16, 3, 8, 9, 16, 3, 26, 3, 15, 16, 8, 3, 19, 6, 10, 9, 9, 3, 24, 9, 20, 9, 6, 3, 45, 3, 8, 12, 9, 9, 26, 3, 13, 9, 24, 3, 24, 3, 6, 12, 11, 9, 24, 3
Offset: 1

Views

Author

Jaroslav Krizek, May 06 2020

Keywords

Comments

Inverse Möbius transform of A009205. - Antti Karttunen, May 19 2020

Examples

			a(6) = gcd(tau(1), sigma(1)) + gcd(tau(2), sigma(2)) + gcd(tau(3), sigma(3)) + gcd(tau(6), sigma(6)) = gcd(1, 1) + gcd(2, 3) + gcd(2, 4) + gcd(4, 12) = 1 + 1 + 2 + 4 = 8.

Links

Crossrefs

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334490 (Sum_{d|n} gcd(d, sigma(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A009205 (gcd(tau(n), sigma(n))).
Cf. A334729 (with product, instead of sum).

Programs

  • Magma
    [&+[GCD(#Divisors(d), &+Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := DivisorSum[n, GCD[DivisorSigma[0, #], DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 07 2020 *)
  • PARI
    a(n) = sumdiv(n, d, gcd(numdiv(d), sigma(d))); \\ Michel Marcus, May 07 2020

Formula

a(p) = 3 for p = odd primes (A065091).

A334731 a(n) = Product_{d|n} gcd(sigma(d), pod(d)) where 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, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 48, 1, 4, 3, 1, 1, 36, 1, 4, 1, 4, 1, 576, 1, 2, 1, 224, 1, 5184, 1, 1, 3, 2, 1, 144, 1, 4, 1, 40, 1, 2304, 1, 16, 9, 4, 1, 2304, 1, 2, 9, 4, 1, 864, 1, 1792, 1, 2, 1, 995328, 1, 4, 1, 1, 1, 20736, 1, 4, 3, 128, 1, 5184, 1, 2
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2020

Keywords

Examples

			a(6) = gcd(sigma(1), pod(1)) * gcd(sigma(2), pod(2)) * gcd(sigma(3), pod(3)) * gcd(sigma(6), pod(6)) = gcd(1, 1) * gcd(3, 2) * gcd(4, 3) * gcd(12, 36) = 1 * 1 * 1 * 12 = 12.

Links

Crossrefs

Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334663 (Sum_{d|n} gcd(sigma(d), pod(d))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A306682 (gcd(sigma(n), pod(n))).

Programs

  • Magma
    [&*[GCD(&+Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := Product[GCD[DivisorSigma[1, d], d^(DivisorSigma[0, d]/2)], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)
  • PARI
    pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(sigma(d[k]), pod(d[k]))); \\ Michel Marcus, May 09-11 2020

Formula

a(p) = 1 for p = primes (A000040).

A334730 a(n) = Product_{d|n} gcd(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, 1, 2, 1, 8, 1, 8, 3, 8, 1, 48, 1, 8, 1, 8, 1, 144, 1, 16, 1, 8, 1, 1536, 1, 8, 3, 16, 1, 256, 1, 16, 1, 8, 1, 7776, 1, 8, 1, 512, 1, 256, 1, 16, 9, 8, 1, 3072, 1, 16, 1, 16, 1, 1152, 1, 512, 1, 8, 1, 36864, 1, 8, 9, 16, 1, 256, 1, 16, 1, 256, 1, 2985984, 1, 8, 3, 16, 1, 256, 1
Offset: 1

Views

Author

Jaroslav Krizek, May 09 2020

Keywords

Examples

			a(6) = gcd(tau(1), pod(1)) * gcd(tau(2), pod(2)) * gcd(tau(3), pod(3)) * gcd(tau(6), pod(6)) = gcd(1, 1) * gcd(2, 2) * gcd(2, 3) * gcd(4, 36) = 1 * 2 * 1 * 4 = 8.

Links

Crossrefs

Cf. A334729 (Product_{d|n} gcd(tau(d), sigma(d))), A334662 (Sum_{d|n} gcd(tau(d), pod(d))).
Cf. A000005 (tau(n)), A007955 (pod(n)), A306671 (gcd(tau(n), pod(n))).

Programs

  • Magma
    [&*[GCD(#Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := Product[GCD[DivisorSigma[0, d], d^(DivisorSigma[0, d]/2)], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 09 2020 *)
  • PARI
    pod(n) = vecprod(divisors(n));
a(n) = my(d=divisors(n)); prod(k=1, #d, gcd(numdiv(d[k]), pod(d[k]))); \\ Michel Marcus, May 09-11 2020

Formula

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

A334806 a(n) = Product_{d|n} lcm(tau(d), sigma(d)) 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, 4, 126, 6, 288, 8, 7560, 156, 1296, 12, 508032, 14, 1152, 576, 1171800, 18, 876096, 20, 1143072, 1024, 2592, 24, 3657830400, 558, 7056, 6240, 4064256, 30, 107495424, 32, 147646800, 2304, 11664, 2304, 1265709908736, 38, 7200, 3136, 24690355200, 42
Offset: 1

Views

Author

Jaroslav Krizek, Jun 26 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 = 288.

Crossrefs

Cf. A334784 (Sum_{d|n} lcm(tau(d), sigma(d))), A334729 (Product_{d|n} gcd(tau(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_] := Product[LCM[DivisorSigma[0, d], DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 41] (* Amiram Eldar, Jun 27 2020 *)
  • PARI
    a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(numdiv(d[k]), sigma(d[k]))); \\ Michel Marcus, Jun 27 2020

Formula

a(p) = p + 1 for p = odd primes (A065091).
Showing 1-4 of 4 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.