A371242 The sum of the unitary divisors of n that are cubefree numbers (A004709).

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

Offset

1,2

Comments

The number of these divisors is A365498(n).

References

D. Suryanarayana, The number and sum of k-free intergers <= 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.

Formula

Multiplicative with a(p^e) = p^e + 1 for e <= 2, and a(p^e) = 1 for e >= 3.

a(n) = 1 if and only if n is cubefull (A036966).

a(n) = A000005(n) if and only if n is squarefree (A005117).

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

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-2)).

Sum_{j=1..n} a(j) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^3 + 1/(p^2 + p)) = 1.16545286600957717104.... .

In general, the formula above holds for the sum of unitary divisors of n that are k-free numbers (k >= 2) with c = Product_{p prime} (1 - 1/p^k + 1/(p^2 + p)) (Suryanarayana, 1969). If k = 2 then c = A065465. In the limit when k -> oo, c = A306633.

Mathematica

f[p_, e_] := If[e < 3, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 3, f[i, 1]^f[i, 2] + 1, 1)); }

Crossrefs

Cf. A000005, A004709, A005117, A034448, A036966, A065465, A092261 (squarefree analog), A306633, A365498.

Sequence in context: A299299 A103402 A154664 * A366743 A191750 A346613

Adjacent sequences: A371239 A371240 A371241 * A371243 A371244 A371245

Keyword

nonn,easy,mult

Author

Amiram Eldar, Mar 16 2024

Status

approved