cp's OEIS Frontend

A381738 Numbers k such that k^2 is abundant.

%I A381738 #12 Mar 06 2025 01:43:56
%S A381738 6,10,12,14,18,20,24,28,30,36,40,42,44,48,50,52,54,56,60,66,68,70,72,
%T A381738 76,78,80,84,88,90,92,96,98,100,102,104,105,108,110,112,114,116,120,
%U A381738 124,126,130,132,136,138,140,144,150,152,154,156,160,162,168,170,174
%N A381738 Numbers k such that k^2 is abundant.
%C A381738 First differs from its subsequence A363171 at n = 21: a(21) = 68 is not a term of A363171.
%C A381738 First differs from its subsequence A334166 at n = 204: a(204) = 585 is not a term of A334166.
%C A381738 A334166 is a subsequence because if k is in A334166, then there is a divisor d of k such that d*k is a Zumkeller number, so d*k is abundant (because all the Zumkeller numbers are abundant), and since d*k is a divisor of k^2 then k^2 is also abundant.
%C A381738 Equivalently, numbers k such that d*k is abundant for at least one divisor d of k.
%C A381738 The least odd term is a(36) = 105.
%C A381738 The least term that is coprime to 6 is a(12519603) = 37182145.
%C A381738 If k is divisible by 6, 10 or 14, then it is a term. Therefore a lower bound for the asymptotic density of this sequence is 19/70 = 0.271... .
%C A381738 The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 33, 347, 3403, 33728, 336599, 3368889, 33628998, 336480309, 3365049432, ... . Apparently, the asymptotic density of this sequence exists and equals 0.336... .
%C A381738 If k is a term then any positive multiple of k is a term. The primitive terms are in A381739.
%H A381738 Amiram Eldar, <a href="/A381738/b381738.txt">Table of n, a(n) for n = 1..10000</a>
%F A381738 a(n) = sqrt(A063734(n)).
%t A381738 Select[Range[200], DivisorSigma[-1, #^2] > 2 &]
%o A381738 (PARI) isok(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2;}
%Y A381738 Cf. A063734.
%Y A381738 Subsequences: A005101, A174830, A334166, A363171, A381739, A381740, A381741, A381742.
%K A381738 nonn,easy
%O A381738 1,1
%A A381738 _Amiram Eldar_, Mar 05 2025