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.

A337324 a(n) is the smallest number m such that gcd(m, tau(m), sigma(m), pod(m)) = 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, 18, 24, 5000, 90, 66339, 56, 288, 3240, 10036224, 60, 582160384, 20412, 16200, 3968, 49030215219, 612, 4637065216, 1520, 142884, 912384, 98881718827959, 480, 7543125, 479232, 3175200, 5824, 26559758051835904, 76950, 25796647321600, 2688, 491774976, 1268973568
Offset: 1

Views

Author

Jaroslav Krizek, Aug 23 2020

Keywords

Comments

From David A. Corneth, Aug 24 2020: (Start)
a(35) <= 1289027059712000000.
a(36) <= 136064563937280.
a(37) = 207816012706349056.
a(38) <= 1835772101525504.
a(39) <= 418089296461824.
a(40) <= 11698803719536640.
gcd(m, tau(m), sigma(m), pod(m)) = gcd(m, tau(m), sigma(m)) which may ease the search.
(End)

Examples

			For n = 6; a(6) = 90 because 90 is the smallest number with gcd(90, tau(90), sigma(90), pod(90)) = gcd(90, 12, 234, 531441000000) = 6.

Crossrefs

Cf. A337323 (gcd(n, tau(n), sigma(n), pod(n))).
Cf. A337325 (least m such that gcd(tau(m), sigma(m), pod(m)) = n).
Cf. A000005, A000203, A007955.

Programs

  • Magma
    [Min([m: m in[1..10^5] | GCD([m, #Divisors(m), &+Divisors(m), &*Divisors(m)]) eq k]): k in [1..10]]
  • PARI
    a(n) = {for(i = 1, oo, f = factor(i); if(gcd([i, numdiv(f), sigma(f)]) == n, return(i)))} \\ David A. Corneth, Aug 24 2020

Extensions

a(11) from Amiram Eldar, Aug 24 2020
More terms from Jaroslav Krizek and David A. Corneth, Aug 24 2020

A329929 a(n) = lcm(tau(n), sigma(n), pod(n)) / gcd(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, 9, 56, 960, 351, 450, 132, 6048, 182, 294, 1800, 158720, 306, 25272, 380, 84000, 14112, 1089, 552, 414720, 11625, 7098, 29160, 32928, 870, 101250, 992, 2064384, 17424, 15606, 58800, 917070336, 1406, 5415, 85176, 11520000, 1722, 777924, 1892
Offset: 1

Views

Author

Jaroslav Krizek, Aug 31 2020

Keywords

Comments

a(n) is also lcm(n, tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n)).

Examples

			a(6) = lcm(tau(6), sigma(6), pod(6)) / gcd(tau(6), sigma(6), pod(6)) = lcm(4, 12, 36) / gcd(4, 12, 36) = 36 / 4 = 9.

Crossrefs

Cf. A336722, A336723, A337323.
Cf. A334985 (lcm(n, tau(n), sigma(n), pod(n)) / gcd(n, tau(n), sigma(n), pod(n))).
Cf. A000005, A000203, A007955.

Programs

  • Magma
    [LCM([#Divisors(n), &+Divisors(n), &*Divisors(n)]) / GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];
  • Mathematica
    a[n_] := LCM @@ (t = {(d = DivisorSigma[0, n]), n^(d/2), DivisorSigma[1, n]}) / GCD @@ t; Array[a, 50] (* Amiram Eldar, Aug 31 2020 *)
  • PARI
    a(n) = my(f=factor(n), v=[numdiv(f), sigma(f), vecprod(divisors(f))]); lcm(v)/gcd(v); \\ Michel Marcus, Aug 31 2020

Formula

a(n) = lcm(n, tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n)).
a(n) = A336723(n) / A336722(n).
a(p) = p * (p+1) for p = primes.

A334985 a(n) = lcm(n, tau(n), sigma(n), pod(n)) / gcd(n, 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, 18, 56, 960, 351, 450, 132, 6048, 182, 588, 1800, 158720, 306, 25272, 380, 84000, 14112, 2178, 552, 414720, 11625, 7098, 29160, 32928, 870, 405000, 992, 2064384, 17424, 15606, 58800, 917070336, 1406, 10830, 85176, 11520000, 1722, 3111696
Offset: 1

Views

Author

Jaroslav Krizek, Sep 22 2020

Keywords

Examples

			a(6) = lcm(6, tau(6), sigma(6), pod(6)) / gcd(6, tau(6), sigma(6), pod(6))  = lcm(6, 4, 12, 36) / gcd(6, 4, 12, 36) = 36 / 2 = 18.

Crossrefs

Cf. A336722, A336723, A337323.
Cf. A329929 (lcm(tau(n), sigma(n), pod(n)) / gcd(tau(n), sigma(n), pod(n))).

Programs

  • Magma
    [LCM([#Divisors(n), &+Divisors(n), &*Divisors(n)]) / GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]]
  • Mathematica
    a[n_] := LCM @@ {(d = DivisorSigma[0, n]), (s = DivisorSigma[1, n]), n^(d/2)} / GCD @@ {n, d, s}; Array[a, 50] (* Amiram Eldar, Sep 22 2020 *)
  • PARI
    a(n) = my(f=factor(n), v=[n, numdiv(f), sigma(f), vecprod(divisors(f))]); lcm(v)/gcd(v); \\ Michel Marcus, Sep 22 2020

Formula

a(n) = lcm(tau(n), sigma(n), pod(n)) / gcd(n, tau(n), sigma(n)).
a(n) = A336723(n) / A337323(n).

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 1-4 of 4 results.

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