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A007433
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Inverse Moebius transform applied twice to squares.
(Formerly M4089)
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10
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1, 6, 11, 27, 27, 66, 51, 112, 102, 162, 123, 297, 171, 306, 297, 453, 291, 612, 363, 729, 561, 738, 531, 1232, 678, 1026, 922, 1377, 843, 1782, 963, 1818, 1353, 1746, 1377, 2754, 1371, 2178, 1881, 3024, 1683
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OFFSET
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1,2
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COMMENTS
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Sum of the squares of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} sigma_2(k)*x^k/(1 - x^k), where sigma_2(k) is the sum of squares of divisors of k (A001157). - Ilya Gutkovskiy, Jan 16 2017
a(n) is multiplicative with a(p^e) = (p^(2*e + 4) - (e+2) * p^2 + e+1)) / (p^2 - 1)^2. - Michael Somos, Jul 15 2018
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EXAMPLE
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G.f. = x + 6*x^2 + 11*x^3 + 27*x^4 + 27*x^5 + 66*x^6 + 51*x^7 + 112*x^8 + 102*x^9 + ... - Michael Somos, Jul 15 2018
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MATHEMATICA
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a[n_] := Plus @@ DivisorSigma[2, Divisors[n]]; Array[a, 41] (* Robert G. Wilson v, May 05 2010 *)
a[ n_] := If[ n < 1, 0, Times @@ (If[ # == 1, 1, (#^(2 #2 + 4) - (#2 + 2) #^2 + #2 + 1) / (#^2 - 1)^2] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 15 2018 *)
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PROG
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a(n)=sumdiv(n, d, sigma(d, 2))
a(n)=sumdiv(n, d, d^2*sigma(n/d, 0))
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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