cp's OEIS Frontend

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.

%I A236288 #7 Jan 24 2014 11:16:34
%S A236288 1,1,4,7,216,144,32768,50625,4826809,34012224,5159780352,481890304,
%T A236288 4049565169664,63403380965376,1521681143169024,25408476896404831,
%U A236288 6746640616477458432,12381557655576425121,13107200000000000000000,53148384174432398229504,38685626227668133590597632
%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.
%C A236288 Conjecture: number 1 is the only number n such that sigma(n)^(n - tau(n)) = sigma(n+1)^(n + 1 - tau(n+1)).
%C A236288 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.
%H A236288 Jaroslav Krizek, <a href="/A236288/b236288.txt">Table of n, a(n) for n = 1..50</a>
%F A236288 a(n) = sigma(n)^(n - tau(n)).
%F A236288 a(n) = A217872(n) / A236287(n) = A000203(n)^n / A000203(n)^A000005(n) = A000203(n)^A049820(n).
%e A236288 a(4) = sigma(4)^(4 - tau(4)) = 7^(4 - 3) = 7.
%t A236288 Table[DivisorSigma[1, n]^[n - DivisorSigma[0, n]], {n, 50}]
%o A236288 (PARI) s=[]; for(n=1, 30, s=concat(s, sigma(n, 1)^(n-sigma(n, 0)))); s \\ _Colin Barker_, Jan 24 2014
%Y A236288 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)).
%K A236288 nonn
%O A236288 1,3
%A A236288 _Jaroslav Krizek_, Jan 23 2014