cp's OEIS Frontend

A328549 1, together with the numbers that are simultaneously superior highly composite (A002201), colossally abundant (A004490), deeply composite (A095848), and miserable average divisor numbers (A263572).

%I A328549 #29 Apr 02 2021 16:30:10
%S A328549 1,2,6,12,60,120,360,2520,5040,55440,720720,1441440
%N A328549 1, together with the numbers that are simultaneously superior highly composite (A002201), colossally abundant (A004490), deeply composite (A095848), and miserable average divisor numbers (A263572).
%C A328549 Presumably there are no further terms.
%C A328549 From _Hal M. Switkay_, Nov 04 2019: (Start)
%C A328549 1. a(n+1) is the product of the first n terms of A328852.
%C A328549 2. This sequence is most rapidly constructed as the intersection of A095849 and A224078. It is designed to list all potential solutions to a question. Let n be a natural number, k real <= 0, e real > 0. Let P(n,k,e) state: on the domain of natural numbers, sigma_k(x)/x^e reaches a maximum at x = n. This implies Q(n,k): sigma_k(n) > sigma_k(m) for m < n a natural number. We ask: for which natural numbers n is it true for all real k <= 0 that there is a real e > 0 such that P(n,k,e)?
%C A328549 If any such n exist, they must belong to the present sequence. A095849 consists of all natural numbers n such that for all real k <= 0, Q(n,k) holds. A224078 consists of all natural numbers n such that for some real e0 and e1 both > 0, P(n,0,e0) and P(n,-1,e1) hold. It would be interesting to see the list of n for which there is an e2 > 0 such that P(n,-2,e2) holds.
%C A328549 Conjecture: the solutions to this problem, if any, form an initial sequence of the present sequence. (End)
%C A328549 Every term of this sequence is also in A065385: a record for the cototient function. - _Hal M. Switkay_, Feb 27 2021
%C A328549 Every term of this sequence, except the first, is also in A210594: factor-dense numbers. - _Hal M. Switkay_, Mar 29 2021
%D A328549 _Hal M. Switkay_, Email to _N. J. A. Sloane_, Oct 20 2019
%Y A328549 1 together with the intersection of A002201, A004490, A095848, A263572.
%Y A328549 Cf. A095849, A224078, A328852, A210594.
%K A328549 nonn,fini,full
%O A328549 1,2
%A A328549 _N. J. A. Sloane_, Oct 20 2019