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A178785 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 such n-perfect number does not exist 0
60, 6552, 222768, 288288, 87360, 49585536, 25486965504, 203558400, 683289600, 556121548800 (list; graph; refs; listen; history; text; internal format)



Let k>=1. In the multiplicative basis Q^(k)={p^(k+1)^j, p runs A000040, j=0,1,...} every positive integer m has unique factorization of the form m=Prod{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 for k tending to infinity, and, by the definition, Q^(infinity)=A000040. Number d is called a k-divisor of m if the exponents d_q in its factorization in 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 n-perfect numbers for n=12,14,15,16 and 18.


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, 1-36.


m = Prod{q is in Q^(k)}q^(m_q) is k-perfect number iff Prod{q is in Q^(k)}(q^((m_q)+1)-1)/(q-1)=2*m.


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.


Cf. A000396, A050376, A007357, A092356.

Sequence in context: A084274 A289307 A091032 * A091753 A130214 A146498

Adjacent sequences:  A178782 A178783 A178784 * A178786 A178787 A178788




Vladimir Shevelev, Jun 14 2010, Jun 18 2010



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Last modified November 21 05:01 EST 2017. Contains 294988 sequences.