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.

A236284 a(n) = tau(n)^n, where tau(n) = A000005(n) = the number of divisors of n.

Original entry on oeis.org

1, 4, 8, 81, 32, 4096, 128, 65536, 19683, 1048576, 2048, 2176782336, 8192, 268435456, 1073741824, 152587890625, 131072, 101559956668416, 524288, 3656158440062976, 4398046511104, 17592186044416, 8388608, 4722366482869645213696, 847288609443, 4503599627370496
Offset: 1

Views

Author

Jaroslav Krizek, Jan 21 2014

Keywords

Examples

			a(4) = tau(4)^4 = 3^4 = 81.

Links

Crossrefs

Cf. A000005 (tau(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236286.

Programs

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

Formula

a(n) = A000005(n)^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.

Links

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).

A236287 a(n) = sigma(n)^tau(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, 9, 16, 343, 36, 20736, 64, 50625, 2197, 104976, 144, 481890304, 196, 331776, 331776, 28629151, 324, 3518743761, 400, 5489031744, 1048576, 1679616, 576, 167961600000000, 29791, 3111696, 2560000, 30840979456, 900, 722204136308736, 1024, 62523502209, 5308416
Offset: 1

Views

Author

Jaroslav Krizek, Jan 23 2014

Keywords

Examples

			a(4) = sigma(4)^tau(4) = 7^3 = 343.

Links

Crossrefs

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236286.

Programs

  • Mathematica
    Table[DivisorSigma[1, n]^DivisorSigma[0, n], {n, 1000}]
  • PARI
    s=[]; for(n=1, 40, s=concat(s, sigma(n, 1)^sigma(n, 0))); s \\ Colin Barker, Jan 24 2014

Formula

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

A236288 a(n) = sigma(n)^n / sigma(n)^tau(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, 1, 4, 7, 216, 144, 32768, 50625, 4826809, 34012224, 5159780352, 481890304, 4049565169664, 63403380965376, 1521681143169024, 25408476896404831, 6746640616477458432, 12381557655576425121, 13107200000000000000000, 53148384174432398229504, 38685626227668133590597632
Offset: 1

Views

Author

Jaroslav Krizek, Jan 23 2014

Keywords

Comments

Conjecture: number 1 is the only number n such that sigma(n)^(n - tau(n)) = sigma(n+1)^(n + 1 - tau(n+1)).
Conjecture: number 1 is the only number n such that sigma(n)^(n - tau(n)) = sigma(k)^(k - tau(k)) has solution for distinct numbers n and k.

Examples

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

Links

Crossrefs

Cf. A000005 (tau(n)), A000203 (sigma(n)), A062758 (n^tau(n)), A217872 (sigma(n)^n), A236285 (tau(n)^sigma(n)), A236287 (sigma(n)^tau(n)).

Programs

  • Mathematica
    Table[DivisorSigma[1, n]^[n - DivisorSigma[0, n]], {n, 50}]
  • PARI
    s=[]; for(n=1, 30, s=concat(s, sigma(n, 1)^(n-sigma(n, 0)))); s \\ Colin Barker, Jan 24 2014

Formula

a(n) = sigma(n)^(n - tau(n)).
a(n) = A217872(n) / A236287(n) = A000203(n)^n / A000203(n)^A000005(n) = A000203(n)^A049820(n).
Showing 1-4 of 4 results.

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