A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90

Offset: 1

Author

Amiram Eldar, Jun 16 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A383763 at n = 32.

The number of these divisors is A365499(n), and the largest of them is A385007(n).

References

• D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Francesco Pappalardi, A survey on k-freeness, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.

Crossrefs

The unitary analog of A385006.

Cf. A036967, A046100, A365499, A385007.

The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), this sequence (biquadratefree).

Programs

Mathematica

f[p_, e_] := If[e < 4, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2] + 1, 1)); }

Formula

Multiplicative with a(p^e) = p^e + 1 for e <= 3, and a(p^e) = 1 for e >= 4.

a(n) = 1 if and only if n is 4-full (A036967).

a(n) <= A034448(n), with equality if and only if n is biquadratefree.

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2) - 1/p^(4*s-3)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2 + p) - 1/p^4) = 1.27769267395905900191... .