A387893 Psi-abundant numbers: numbers k such that psi(k) > 2*k.

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 180, 186, 198, 204, 210, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282, 294, 300, 306, 312, 318, 330, 336, 342, 348, 350, 354, 360, 366, 372, 378, 390, 396, 402, 408, 414, 420, 426, 438, 444, 450

Offset

1,1

Comments

From Amiram Eldar, Sep 11 2025: (Start)

A squarefree number is a term if and only if it is abundant (i.e., in A087248).

A number is a term if and only if its squarefree kernel (A007947) is abundant (i.e., in A087248).

If p_1, p_2, ..., p_k are the distinct prime factors of a number, then it is a term if and only if Product_{i=1..k} (1 + 1/p_i) > 2. (End)

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 8, 152, 1758, 17895, 180135, 1801173, 18009259, 180141194, 1801422258, 18012950712, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1801... . - Amiram Eldar, Nov 10 2025

Links

Table of n, a(n) for n=1..61.

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Example

30 is in the sequence since psi(30) = 72 > 60.

Mathematica

psi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}]; Select[Range[450], psi[#]>2# &] (* Stefano Spezia, Sep 11 2025 *)

Prog

(PARI) isok(k) = vecprod(apply(x -> 1 + 1/x, factor(k)[, 1])) > 2; \\ Amiram Eldar, Sep 11 2025

Crossrefs

Cf. A001615, A005101, A007947, A033845 (psi-perfect), A087248, A387892 (psi-deficient).

Sequence in context: A000977 A214195 A033992 * A360525 A308127 A390558

Adjacent sequences: A387890 A387891 A387892 * A387894 A387895 A387896

Keyword

nonn,easy

Author

S. I. Dimitrov, Sep 11 2025

Status

approved