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.

A236285 a(n) = tau(n)^sigma(n), where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 8, 16, 2187, 64, 16777216, 256, 1073741824, 1594323, 68719476736, 4096, 6140942214464815497216, 16384, 281474976710656, 281474976710656, 4656612873077392578125, 262144, 2227915756473955677973140996096, 1048576, 481229803398374426442198455156736
Offset: 1

Views

Author

Jaroslav Krizek, Jan 21 2014

Keywords

Examples

			a(4) = tau(4)^sigma(4) = 3^7 = 2187.
		

Crossrefs

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236284 (tau(n)^n), A236286.

Programs

  • Mathematica
    Table[DivisorSigma[0, n]^DivisorSigma[1, n], {n, 1000}]
  • PARI
    s=[]; for(n=1, 20, s=concat(s, sigma(n, 0)^sigma(n))); s \\ Colin Barker, Jan 22 2014

Formula

a(n) = A000005(n)^A000203(n).

A236286 a(n) = tau(n)^sigma(n) / tau(n)^n, where tau(n) = A000005(n) = the number of divisors of n and sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

1, 2, 2, 27, 2, 4096, 2, 16384, 81, 65536, 2, 2821109907456, 2, 1048576, 262144, 30517578125, 2, 21936950640377856, 2, 131621703842267136, 4194304, 268435456, 2, 324518553658426726783156020576256, 729, 4294967296, 67108864, 6140942214464815497216, 2
Offset: 1

Views

Author

Jaroslav Krizek, Jan 21 2014

Keywords

Comments

a(n) = tau(n)^sigma_p(n), where sigma_p(n) = A001065(n) = the sum of proper divisors of n.

Examples

			a(4) = tau(4)^sigma(4) / tau(4)^4 = 3^7 /3^4 = 27.
		

Crossrefs

Cf. A000005 (tau(n)), A000203 (sigma(n)), A001065 (sigma_p(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236284 (tau(n)^n), A236285 (tau(n)^sigma(n)).

Programs

  • Mathematica
    Table[DivisorSigma[0, n]^[DivisorSigma[1, n] - n], {n, 1000}]
  • PARI
    s=[]; for(n=1, 30, s=concat(s, sigma(n, 0)^sigma(n)/sigma(n, 0)^n)); s \\ Colin Barker, Jan 22 2014

Formula

a(n) = A236285(n) / A236284(n) = A000005(n)^A000203(n) / A000005(n)^n = A000005(n)^A001065(n).
a(p) = 2 for p = primes (A000040).

A331667 Numbers m with a divisor d such that tau(d)^d = m.

Original entry on oeis.org

1, 4, 19683, 65536, 2176782336, 101559956668416, 7958661109946400884391936
Offset: 1

Views

Author

Jaroslav Krizek, Feb 28 2020

Keywords

Comments

Corresponding values of divisors d and tau(d) are 1, 2, 9, 8, 12, 18, 32, ..., and 1, 2, 3, 4, 6, 6, 6, ..., respectively.
a(8) = 2^144*3^72 = 12^72 is too large to be included in Data. - Giovanni Resta, Feb 28 2020

Examples

			19683 is a term because 3^9 = 19683; 9 divides 19683, tau(9) = 3.
		

Crossrefs

Cf. A236284 (tau(n)^n).

Programs

  • Magma
    [n: n in [1..100000] | #[d: d in Divisors(n) | NumberOfDivisors(d)^d eq n] ge 1]
    
  • Mathematica
    seqQ[n_] := AnyTrue[Divisors[n], DivisorSigma[0, #]^# == n &]; Select[Range[70000], seqQ] (* Amiram Eldar, Feb 28 2020 *)
  • PARI
    isok(m) = fordiv(m, d, if (numdiv(d)^d == m, return (1));); \\ Michel Marcus, Mar 07 2020

Extensions

More terms from Giovanni Resta, Feb 28 2020
Showing 1-3 of 3 results.