

A095848


Deeply composite numbers: numbers n where sigma_k(n) increases to a record for all sufficiently low (i.e., negative) values of k.


4



1, 2, 4, 6, 12, 24, 48, 60, 120, 240, 360, 420, 840, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 360360, 720720, 1441440, 2162160, 3603600, 7207200, 10810800, 12252240, 24504480, 36756720, 61261200, 122522400
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OFFSET

1,2


COMMENTS

Sigma_k(n) > sigma_k(m) for all m < n (where the function sigma_k(n) is the sum of the kth powers of all divisors of n) for all or almost all negative values of k.


LINKS

T. D. Noe, Table of n, a(n) for n=1..448 (terms < 10^100)
Wikipedia, Table of divisors.


FORMULA

For n>=4, a(n) is the smallest integer > a(n1) such that the list of its divisors precedes the list of a(n1)'s divisors in lexicographic order.


EXAMPLE

The list of the divisors of a(6)=24, {1,2,3,4,6,8,12,24}, lexicographically precedes the list for the previous term in the sequence (in this case, {1,2,3,4,6,12}, the list for a(5)=12). Therefore 24 belongs in the sequence. 36 does not satisfy this requirement, as {1,2,3,4,6,9 . . .} comes after {1,2,3,4,6,8 . . .} in lexicographic order. Since 8^k/9^k increases without limit as k decreases, sigma(36)_k < sigma(24)_k for almost all negative values of k; therefore 36 does not belong in the sequence.


CROSSREFS

Cf. A004394, A095849.
Sequence in context: A048115 A047151 A068010 * A208767 A136339 A019505
Adjacent sequences: A095845 A095846 A095847 * A095849 A095850 A095851


KEYWORD

nonn


AUTHOR

Matthew Vandermast, Jun 09 2004


STATUS

approved



