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.

1, 1, 4, 7, 216, 144, 32768, 50625, 4826809, 34012224, 5159780352, 481890304, 4049565169664, 63403380965376, 1521681143169024, 25408476896404831, 6746640616477458432, 12381557655576425121, 13107200000000000000000, 53148384174432398229504, 38685626227668133590597632

Offset

1,3

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.

Links

Jaroslav Krizek, Table of n, a(n) for n = 1..50

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).

Example

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

Mathematica

Table[DivisorSigma[1, n]^[n - DivisorSigma[0, n]], {n, 50}]

Prog

(PARI) s=[]; for(n=1, 30, s=concat(s, sigma(n, 1)^(n-sigma(n, 0)))); s \\ Colin Barker, Jan 24 2014

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)).

Sequence in context: A362857 A289381 A126577 * A348131 A073164 A297215

Adjacent sequences: A236285 A236286 A236287 * A236289 A236290 A236291

Keyword

nonn

Author

Jaroslav Krizek, Jan 23 2014

Status

approved