

A178785


a(n) is the smallest nperfect number of the form 2^(n+1)*L, where L is an odd number with exponents <= n in its prime power factorization, and a(n)=0 if no such nperfect number exists.


0




OFFSET

1,1


COMMENTS

Let k >= 1. In the multiplicative basis Q^(k) = {p^(k+1)^j, p runs through A000040, j=0,1,...} every positive integer m has a unique factorization of the form m = Product_{q is in Q^(k)} q^(m_q), where m_q is in {0,1,...,k}. In particular, in the case of k=1, we have the unique factorization over distinct terms of A050376. Notice that the standard prime basis is the limiting value for k tending to infinity, and, by the definition, Q^(infinity)=A000040. The number d is called a kdivisor of m if the exponents d_q in its factorization in the basis Q^(k) do not exceed m_q. A number m is called kperfect if it equals to the sum of its proper positive kdivisors. Conjecture: a(11)=0. Note that we also know of nperfect numbers for n = 12, 14, 15, 16, and 18.


LINKS

Table of n, a(n) for n=1..10.
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 136.


FORMULA

m = Product_{q is in Q^(k)} q^(m_q) is a kperfect number iff Product_{q is in Q^(k)} (q^((m_q)+1)1)/(q1) = 2*m.


EXAMPLE

In case of n=2, we have the basis ("2primes"): 2,3,5,7,8,11,13,... By the formula, we construct from the left m and from the right 2*m. By the condition, m begins from "2prime" 8. From the right we have 8+1=3^2, therefore from the left we have 8*3^2 and from the right 3^2*(3^31)/(31)=3^2*13. Thus from the left it should be 8*3^2*13 and from the right 3^2*13*14. Finally, from the left we obtain m=8*3^2*13*7=6552 and from the right we have 2*m=3^2*13*14*8. By the construction, it is the smallest 2perfect number of the required form. Thus a(2)=6552.


CROSSREFS

Cf. A000396, A050376, A007357, A092356.
Sequence in context: A289307 A091032 A328951 * A091753 A336629 A303790
Adjacent sequences: A178782 A178783 A178784 * A178786 A178787 A178788


KEYWORD

nonn,more


AUTHOR

Vladimir Shevelev, Jun 14 2010, Jun 18 2010


STATUS

approved



