A380090 The sum of the unitary divisors of n that are terms in A207481.

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, 28, 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, 84, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96

Offset: 1

Author

Amiram Eldar, Jan 12 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A371242 at n = 27.

Links

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

Crossrefs

Cf. A005117, A054743, A077610, A185359, A207481, A371242, A377520, A380087, A380088, A380089.

Programs

Mathematica

f[p_, e_] := If[e <= p, p^e, 0] + 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] <= f[i,1], f[i,1]^f[i,2], 0) + 1);}

Formula

a(n) = A034448(A380088(n)).

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

a(n) = 1 if and only if n is in A054743.

a(n) < A034448(n) if and only if n is in A185359.

a(n) = A034448(n) if and only if n is in A207481.

a(n) = A377520(n) if and only if n is squarefree (A005117).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(p+2) + p^(p+1) + p^p - p - 1)/(p^(p+1) * (p+1)) = 1.2078161... .