|
| |
|
|
A336219
|
|
a(n) is the smallest member of the n-th purely periodic unitary sigma aliquot cycle listed in A336216.
|
|
2
|
|
|
|
6, 30, 60, 90, 114, 1140, 1482, 2418, 18018, 24180, 32130, 35238, 44772, 56430, 67158, 87360, 142310, 180180, 197340, 241110, 263820, 296010, 308220, 395730, 462330, 473298, 591030, 669900, 671580, 698130, 763620, 785148, 815100, 1004850, 1077890, 1080150, 1156870, 1177722
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
This is the first column of the irregular triangle in A336216.
From the formula of Vladeta Jovovic in A034448 we get for an even number n not divisible by 4 and odd prime p: usigma(2^m * p * n) = (2^(m+1) + 1) * (p + 1) * usigma(n) / 3 so that usigma(2^m * p * n) = (2^m * p * n) * usigma(n) when 3* 2^m * p = (2^(m+1) + 1) * (p + 1), and consequently, p = (2^(m+1) + 1) / (2^m - 1), i.e. p = 5 for m = 1, and p = 3 for m = 2.
Therefore, if all members a_1, a_2, ... , a_k, a_1 of a cycle are even and not divisible by 4 and 5 then 10*a_1, 10*a_2, ... , 10*a_k, 10*a_1 form a cycle, and if all members a_1, a_2, ... , a_k, a_1 of a cycle are even and not divisible by 3 and 4 then 12*a_1, 12*a_2, ... , 12*a_k, 12*a_1 form a cycle.
If all members a_1, a_2, ... , a_k, a_1 of a cycle are odd, divisible by 3, but not divisible by 5 and 9 then 15*a_1, 15*a_2, ... , 15*a_k, 15*a_1 form a cycle. No such cycles exist in the current data up to 27287260.
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
Start numbers of cycles related by a factor of 10 or 12, respectively:
10: (6, 60), (114, 1140), (2418, 24180), (18018, 180180), (67158, 671580), (1177722, 1777220), ...
12: (142310, 1707720), (1077890, 12934680), (1156870, 13882440), (1475810, 17709720), ...
|
|
|
MATHEMATICA
|
(* a336216 and support functions in A336216 *)
Map[First, a336216[100000]] (* a(1..16) *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
