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

Author

Amiram Eldar, Jun 16 2025

Keywords

• nonn
• easy
• mult

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

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).

Programs

Mathematica

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

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... .