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Unitary Divisor Function


UnitaryDivisorFunctions

The unitary divisor function sigma_k^*(n) is the analog of the divisor function sigma_k(n) for unitary divisors and denotes the sum-of-kth-powers-of-the-unitary divisors function. As in the case of the usual divisor function, sigma_1^*(n) is commonly written sigma^*(n).

The numbers of unitary divisors sigma_0^*(n) is the same as the numbers of squarefree divisors of n, as well as 2^q, where q is the number of different primes dividing n.

If n is squarefree, then sigma(n)=sigma^*(n).

sigma_k^*(n) can be computed using the formula

 sigma_k^*(p_1^(alpha_1)p_2^(alpha_2)...)=(1+p_1^(kalpha_1))(1+p_2^(kalpha_2))...,

which can be computed in the Wolfram Language as

  UnitaryDivisorSigma[k_, n_Integer] := Times @@
    (1 + (Power @@@ FactorInteger[n])^k)

The following table gives sigma_k^*(n) for n=1, 2, ... and small k.

kOEISsigma_k^*(n) for n=1, 2, ...
0A0344441, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, ...
1A0344481, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, ...
2A0346761, 5, 10, 17, 26, 50, 50, 65, 82, 130, 122, 170, 170, 250, 260, ...
3A0346771, 9, 28, 65, 126, 252, 344, 513, 730, 1134, 1332, 1820, ...
4A0346781, 17, 82, 257, 626, 1394, 2402, 4097, 6562, 10642, ...

See also

Divisor Function, Unitary Amicable Pair, Unitary Divisor

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References

Sloane, N. J. A. Sequences A034444, A034448, A034676, A034677, and A034678 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Unitary Divisor Function

Cite this as:

Weisstein, Eric W. "Unitary Divisor Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UnitaryDivisorFunction.html

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