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A322485
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The sum of the semi-unitary divisors of n.
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5
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1, 3, 4, 5, 6, 12, 8, 11, 10, 18, 12, 20, 14, 24, 24, 19, 18, 30, 20, 30, 32, 36, 24, 44, 26, 42, 31, 40, 30, 72, 32, 39, 48, 54, 48, 50, 38, 60, 56, 66, 42, 96, 44, 60, 60, 72, 48, 76, 50, 78, 72, 70, 54, 93, 72, 88, 80, 90, 60, 120, 62, 96, 80, 71, 84, 144
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OFFSET
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1,2
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COMMENTS
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A semi-unitary divisor of n is defined as the largest divisor d of n such that the largest divisor of d that is a unitary divisor of n/d is 1 (see A322483).
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REFERENCES
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J. Chidambaraswamy, Sum functions of unitary and semi-unitary divisors, J. Indian Math. Soc., Vol. 31 (1967), pp. 117-126.
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LINKS
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FORMULA
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Multiplicative with a(p^e) = sigma(p^floor((e-1)/2)) + p^e = (p^floor((e+1)/2) - 1)/(p-1) + p^e.
In particular a(p) = p + 1, a(p^2) = p^2 + 1, a(p^3) = p^3 + p + 1.
a(n) <= A000203(n) with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(3)/2) * Product_{p prime} (1 - 2/p^3 + 1/p^5) = 0.7004703314... . - Amiram Eldar, Nov 24 2022
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EXAMPLE
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The semi-unitary divisors of 8 are 1, 2, 8 (4 is not semi-unitary divisor since the largest divisor of 4 that is a unitary divisor of 8/4 = 2 is 2 > 1), and their sum is 11, thus a(8) = 11.
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MATHEMATICA
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f[p_, e_] := (p^Floor[(e+1)/2] - 1)/(p-1) + p^e; susigma[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Array[susigma, 100]
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PROG
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(PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, my(p=f[k, 1], e=f[k, 2]); f[k, 1] = (p^((e+1)\2) - 1)/(p-1) + p^e; f[k, 2] = 1; ); factorback(f); } \\ Michel Marcus, Dec 14 2018
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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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STATUS
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approved
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