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COMMENT
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If n=Product p_i^r_i then the unitary ordinary sigma function is UO-sigma(n)= UnitarySigma(2^r_1)*Sigma(n/2^r_1) =(2^r_1+1)*Product(p_i^(r_i+1)-1)/(p_i-1), where p_i is not 2.
The initial values of k are 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2. However, I conjecture that every positive integer >= 2 must appear.
E.g. UO-sigma(2^4*7^2)=UnitarySigma(2^4)*sigma(7^2)=17*57= 969. So UO-sigma(n) = UnitarySigma(n) if n=2^r, or = sigma(n) if GCD(2,n)=1. A UO-sigma perfect number satisfies UO-sigma(n) = k*n for some k.
Some interesting subsequences exist: s(n) := {a(1), a(4), a(9), a(11), ...} has the property that s(n-1)|s(n): 2*3, 2^3*3^2*7*13, 2^5*3^2*7*13*11, 2^7*3^2*7*11*13*43, 2^8*3^2*7*11*13*43*257,
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EXAMPLE
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Sequence begins 2*3, 2^2*3*5, 2^3*3^3*5, 2^3*3^2*7*13,
2^4*3^3*5*17, 2^5*3^3*5*11, 2^6*3*5*7*13, 2^4*3^2*7*13*17, 2^5*3^2*7*13*11,
2^6*3^2*5*7*13^2*31*61, 2^7*3^2*7*11*13*43, 2^7*3^3*5*11*43, ...
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