A254001 - OEIS

%I #16 Jan 25 2015 22:13:47

%S 0,0,0,0,0,1,0,0,0,2,0,1,0,2,0,0,0,1,0,1,0,3,0,1,0,3,0,1,0,1,0,0,0,4,

%T 0,1,0,4,0,1,0,1,0,2,0,4,0,1,0,2,0,2,0,1,0,1,0,4,0,1,0,4,0,0,0,1,0,2,

%U 0,1,0,1,0,5,0,2,0,1,0,1,0,5,0,1,0,5,0,1,0,1,0,2,0,5,0,1,0,2,0,1

%N a(n) is the least natural number k such that n^k is abundant or perfect, or a(n) is 0 if all n^k are deficient numbers.

%C Let p_1, p_2, ..., p_m be the distinct primes dividing n. If (p_1/(p_1 - 1))*(p_2/(p_2 - 1))*...*(p_m/(p_m - 1)) > 2, then sufficiently high powers of n are abundant. Otherwise all powers of n are deficient, and we set a(n)=0.

%C The sequence is unbounded. In particular, for each N, we have a(A063765(N)) = N.

%H Jeppe Stig Nielsen, <a href="/A254001/b254001.txt">Table of n, a(n) for n = 1..10000</a>

%e a(38)=4 because 38^4 is abundant (A023196) while 38^3, 38^2, and 38 are all deficient (A005100).

%o (PARI) a(n) = {

%o primeVect = factor(n)[,1];

%o if(prod(i=1,#primeVect,1-1/primeVect[i])>=1/2,return(0));

%o for(k=1,10^99,t=n^k;if(sigma(t)>=2*t,return(k))); }

%K nonn,easy

%O 1,10

%A _Jeppe Stig Nielsen_, Jan 22 2015