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A328549
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1, together with the numbers that are simultaneously superior highly composite (A002201), colossally abundant (A004490), deeply composite (A095848), and miserable average divisor numbers (A263572).
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1
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1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Presumably there are no further terms.
1. a(n+1) is the product of the first n terms of A328852.
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)?
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.
Conjecture: the solutions to this problem, if any, form an initial sequence of the present sequence. (End)
Every term of this sequence is also in A065385: a record for the cototient function. - Hal M. Switkay, Feb 27 2021
Every term of this sequence, except the first, is also in A210594: factor-dense numbers. - Hal M. Switkay, Mar 29 2021
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REFERENCES
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LINKS
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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