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%I #17 Apr 29 2021 02:23:48
%S 60,6552,222768,288288,87360,49585536,25486965504,203558400,683289600,
%T 556121548800
%N a(n) is the smallest n-perfect 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 n-perfect number exists.
%C 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 k-divisor 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 k-perfect if it equals to the sum of its proper positive k-divisors. Conjecture: a(11)=0. Note that we also know of n-perfect numbers for n = 12, 14, 15, 16, and 18.
%H S. Litsyn and V. S. Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/h33/h33.Abstract.html">On factorization of integers with restrictions on the exponent</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.
%F m = Product_{q is in Q^(k)} q^(m_q) is a k-perfect number iff Product_{q is in Q^(k)} (q^((m_q)+1)-1)/(q-1) = 2*m.
%e In case of n=2, we have the basis ("2-primes"): 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 "2-prime" 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*(3^3-1)/(3-1)=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 2-perfect number of the required form. Thus a(2)=6552.
%Y Cf. A000396, A050376, A007357, A092356.
%K nonn,more
%O 1,1
%A _Vladimir Shevelev_, Jun 14 2010, Jun 18 2010
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