A380087 The sum of the unitary divisors of n that are terms in A276078.

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

Offset

1,2

Links

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

Formula

a(n) = A034448(A380085(n)).

Multiplicative with a(p^e) = p^e + 1 if e <= pi(p) = A000720(p), and 1 otherwise.

a(n) = 1 if and only if n is in A325127.

a(n) < A034448(n) if and only if n is in A276079.

a(n) = A034448(n) if and only if n is in A276078.

a(n) = A377517(n) if and only if n is squarefree (A005117).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(k+2) + p(k)^(k+1) + p(k)^k - p(k) - 1)/(p(k)^(k+1) * (p(k)+1)) = 1.03676..., where p(k) = prime(k).

Mathematica

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

Crossrefs

Cf. A000720, A005117, A034448, A077610, A276078, A377517, A380085, A380086, A380090.

Sequence in context: A281862 A366795 A092261 * A389450 A389451 A367991

Adjacent sequences: A380084 A380085 A380086 * A380088 A380089 A380090

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 11 2025

Status

approved