A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 26, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 50, 26, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38

Offset

1,5

Comments

First differs from A186099 at n = 25; a(25) = 26, while A186099(25) = 31.

The number of these divisors is A385044(n), and the largest of them is A065330(n).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = 1 if p <= 3, and p^e + 1 if p >= 5.

a(n) = A034448(n)/A385046(n).

a(n) <= A034448(n), with equality if and only if n is 5-rough.

a(n) <= A186099(n).

Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1)) * ((1-1/2^(s-1))/(1-1/2^(2*s-1))) * ((1-1/3^(s-1))/(1-1/3^(2*s-1))).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/(91*zeta(3)) = 0.270679... .

Mathematica

f[p_, e_] := If[p <= 3, 1, p^e + 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, 1] < 5, 1, f[i, 1]^f[i, 2] + 1)); }

Crossrefs

The unitary analog of A186099.

Cf. A002117, A007310, A034448, A065330, A385044.

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), this sequence (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

Sequence in context: A156921 A094214 A001622 * A186099 A021622 A073228

Adjacent sequences: A385042 A385043 A385044 * A385046 A385047 A385048

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 16 2025

Status

approved