A254001 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.

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, 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, 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

Offset

1,10

Comments

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.

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

Links

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000

Example

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

Prog

(PARI) a(n) = {
primeVect = factor(n)[, 1];
if(prod(i=1, #primeVect, 1-1/primeVect[i])>=1/2, return(0));
for(k=1, 10^99, t=n^k; if(sigma(t)>=2*t, return(k))); }

Crossrefs

Sequence in context: A108069 A227837 A263099 * A089734 A321375 A352555

Adjacent sequences: A253998 A253999 A254000 * A254002 A254003 A254004

Keyword

nonn,easy

Author

Jeppe Stig Nielsen, Jan 22 2015

Status

approved