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.
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)
a(38) = 1412875743840099 is the next term after 9 not divisible by 15. The number of terms not divisible by 7 is conjectured to be infinite provided no odd perfect numbers exist. - Alexander Violette, Sep 26 2025
LINKS
Max Alekseyev and Alexander Violette, Table of n, a(n) for n = 1..55 (terms 1..31 from Giovanni Resta)
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