A383697 Exponential squarefree exponential abundant numbers: numbers k such that A361174(k) > 2*k.

900, 1764, 4356, 4500, 4900, 6084, 6300, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 36900, 38700, 40572, 42300, 42588, 44100, 47700, 47916, 49284, 49500

Offset

1,1

Comments

Subsequence of A383693 and first differs from it at n = 21.

All the terms are nonsquarefree numbers (A013929), since A361174(k) = k if k is a squarefree number (A005117).

The least odd term is a(198045) = 225450225, and the least term that is coprime to 6 is a(9.815...*10^17) = 1117347505588495206025.

The least term that is not an exponentially squarefree number (A209061) is a(8.85...*10^1324) = 2^4 * Product_{k=2..248} prime(k)^2 = 1.00786...*10^1328.

The asymptotic density of this sequence is Sum_{n>=1} f(A383698(n)) = 0.000878475..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)).

Links

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

Mathematica

f[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; q[k_] := Times @@ f @@@ FactorInteger[k] > 2*k; Select[Range[1000], q]

Prog

(PARI) ff(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
isok(k) = {my(f = factor(k)); prod(i=1, #f~, ff(f[i, 1], f[i, 2])) > 2*k; }

Crossrefs

Subsequence of A013929, A129575 and A383693.

A383698 is a subsequence.

Cf. A005117, A209061, A361174.

Sequence in context: A321206 A336680 A383693 * A391283 A383694 A383698

Adjacent sequences: A383694 A383695 A383696 * A383698 A383699 A383700

Keyword

nonn,easy

Author

Amiram Eldar, May 06 2025

Status

approved