A RECURSION FOR DIVISOR FUNCTION OVER DIVISORS BELONGING TO A PRESCRIBED FINITE SEQUENCE OF POSITIVE INTEGERS AND A SOLUTION OF THE LAHIRI PROBLEM FOR DIVISOR FUNCTION σx(n)
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چکیده
(2) σ(n) = σ(n−1)+σ(n−2)−σ(n−5)−σ(n−7)+σ(n−12)+σ(n−15)−... where the numbers 1,2,5,7,12,15,... appearing in the successive terms in (1)-(2) are the positive pentagonal numbers {vm} given by (3) vm = m(3m∓ 1)/2, m = 1, 2, ... In identities (1)-(2) we accept that p(m) = 0, σ(m) = 0 when m < 0. The only formal difference is that (1) is true with the understanding that (4) p(0) = 1, while (2) is valid with the understanding that (5) σ(0) = n. Note that, formulas (1)-(2) are proved with help of the famous Euler pentagonal identity
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تاریخ انتشار 2009