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A361206
Lexicographically earliest infinite sequence of distinct imperfect numbers such that the sum of the abundance of all terms is never < 1.
0
12, 1, 2, 4, 18, 3, 8, 20, 10, 24, 5, 7, 16, 30, 9, 14, 32, 36, 11, 13, 40, 15, 42, 17, 48, 19, 21, 54, 22, 44, 56, 50, 60, 23, 25, 52, 64, 66, 26, 70, 72, 27, 29, 34, 78, 45, 80, 33, 68, 84, 31, 35, 88, 90, 37, 38, 96, 39, 41, 100, 46, 102, 76, 104, 108, 43, 58
OFFSET
1,1
COMMENTS
The abundance of each term is A033880(a(n)) and s = Sum_{i=1..n} A033880(a(i)).
All imperfect numbers A132999 will appear in this sequence and the abundant numbers A005101 will appear in order.
EXAMPLE
The sequence starts with a(1) = 12, since 12 is the first imperfect number with abundance greater than 0. Then the next term not yet in the sequence, such that s is not less than 1, is 1.
a(5) is the next abundant number 18, since any deficient number would bring s below 1.
n : 1 2 3 4 5 6 7 8 9 10
a(n): 12 1 2 4 18 3 8 20 10 24
s : 4 3 2 1 4 2 1 3 1 13
PROG
(Python)
from sympy.ntheory import abundance
from itertools import count, filterfalse
def A361206_list(nmax):
A, s = [], 0
for n in range(1, nmax+1):
A2 = set(A)
for y in filterfalse(A2.__contains__, count(1)):
ab = abundance(y)
if ab != 0 and ab + s >= 1:
A.append(y)
s += ab
break
return(A)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John Tyler Rascoe, Mar 04 2023
STATUS
approved