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

Author

Jaroslav Krizek, Jan 23 2014

Keywords

• nonn

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

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

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).