OFFSET
1,2
COMMENTS
The abundancy of a number k is defined as A(k) = sigma(k)/k. Deficient numbers have an abundancy less than 2. This sequence has terms in common with A171929. Sequence A188263, which deals with abundant numbers, approaches 2 from above. The similar sequence for even numbers consists of the powers of 2.
a(29) > 10^12. - Donovan Johnson, Apr 08 2011
This sequence is finite iff there is an odd perfect number (which would have abundancy 2). Otherwise, one always has a subsequent term a(n+1) <= a(n)*p where p is the smallest prime not dividing a(n) and larger than 1/(2/A(a(n))-1). Indeed, such an a(n)*p is still deficient but has abundancy larger than a(n), thus closer to 2. - M. F. Hasler, Feb 22 2017
From M. F. Hasler, Jan 25 2020: (Start)
The upper bounds a(n)*p mentioned above are often terms of the sequence, but not the subsequent but a later one: e.g., 9*5 = 45, 15*7 = 105, 45*7 = 315, 105*11 = 1155, 315*107 = 33705, 1155*389 = 449295, 26325*389 = 10240425, ...
Is 9 the largest term not divisible by 15? Only 7 of the 26 terms listed after 45 are not multiples of 7: is this subsequence finite? (End)
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..31 (terms < 10^13)
MATHEMATICA
k = 1; minDiff = 1; Join[{k}, Table[k = k + 2; While[abun = DivisorSigma[1, k]/k; 2 - abun > minDiff || abun => 2, k = k + 2]; minDiff = 2 - abun; k, {10}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 05 2011
EXTENSIONS
a(22)-a(28) from Donovan Johnson, Apr 08 2011
STATUS
approved