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.

A334662 a(n) = Sum_{d|n} gcd(tau(d), pod(d)), where pod(k) is the product of the divisors of k (A007955).

Original entry on oeis.org

1, 3, 2, 4, 2, 8, 2, 8, 5, 8, 2, 15, 2, 8, 4, 9, 2, 17, 2, 11, 4, 8, 2, 27, 3, 8, 6, 11, 2, 22, 2, 11, 4, 8, 4, 33, 2, 8, 4, 23, 2, 22, 2, 11, 10, 8, 2, 30, 3, 11, 4, 11, 2, 26, 4, 23, 4, 8, 2, 43, 2, 8, 10, 12, 4, 22, 2, 11, 4, 22, 2, 57, 2, 8, 8, 11, 4, 22
Offset: 1

Views

Author

Jaroslav Krizek, May 07 2020

Keywords

Comments

Inverse Möbius transform of A306671. - Antti Karttunen, May 19 2020

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. A334579 (Sum_{d|n} gcd(tau(d), sigma(d))), A334663 (Sum_{d|n} gcd(sigma(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]]
  • PARI
    a(n) = sumdiv(n, d, gcd(numdiv(d), vecprod(divisors(d)))); \\ Michel Marcus, May 08 2020

Formula

a(p) = 2 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).

A334794 a(n) = Sum_{d|n} lcm(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, 7, 13, 63, 31, 55, 57, 1023, 364, 937, 133, 12207, 183, 1239, 1843, 32767, 307, 76222, 381, 168993, 14181, 4495, 553, 1672047, 3906, 14385, 29524, 23247, 871, 812785, 993, 2097151, 17569, 31525, 58887, 917158710, 1407, 22047, 85371, 23209953, 1723, 6238791
Offset: 1

Views

Author

Jaroslav Krizek, May 12 2020

Keywords

Examples

			a(6) = lcm(sigma(1), pod(1)) + lcm(sigma(2), pod(2)) + lcm(sigma(3), pod(3)) + lcm(sigma(6), pod(6)) = lcm(1, 1) + lcm(3, 2) + lcm(4, 3) + lcm(12, 36) = 1 + 6 + 12 + 36 = 55.

Crossrefs

Cf. A334663 (Sum_{d|n} gcd(sigma(d), pod(d))), A334793 (Sum_{d|n} lcm(tau(d), pod(d))).
Cf. A000203 (sigma(n)), A007955 (pod(n)), A324529 (lcm(sigma(n), pod(n))).

Programs

  • Magma
    [&+[LCM(&+Divisors(d), &*Divisors(d)): d in Divisors(n)]: n in [1..100]]
  • Mathematica
    a[n_] := DivisorSum[n, LCM[DivisorSigma[1, #], #^(DivisorSigma[0, #]/2)] &]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)
  • PARI
    a(n) = sumdiv(n, d, lcm(sigma(d), vecprod(divisors(d)))); \\ Michel Marcus, May 12 2020

Formula

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

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