A336219 a(n) is the smallest member of the n-th purely periodic unitary sigma aliquot cycle listed in A336216.

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

Offset: 1

Author

Hartmut F. W. Hoft, Jul 12 2020

Keywords

• nonn

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.

Examples

			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), ...
		

Crossrefs

Cf. A034448, A063919, A327157, A336216, A336218.

Programs

Mathematica

(* a336216 and support functions in A336216 *)
Map[First, a336216[100000]] (* a(1..16) *)

Formula

a(n) = A336216( 1 + Sum_{i=1..n-1} A336218(i) ).