A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

3, 9, 10, 44, 70, 102, 350, 372, 1608, 3492, 6096, 10380, 44040, 100260, 180240, 425160, 1744560, 2425080, 5509980, 10048080, 23614920, 97639920, 396315360, 900229680, 2519017200, 3113704440, 12870562320, 52307529120

Offset

1,1

Comments

None of the terms are superabundant (A004394) since all the superabundant numbers are practical numbers (A005153).

The least term m that is k-abundant (having sigma(m)/m > k) for k = 2, 3, ... is A005101(14) = 70, A068403(896) = 44040, A068404(792087) = 3113704440, ...

What is the least 5-abundant number (A215264) that is not practical?

Links

Table of n, a(n) for n=1..28.

Example

The first 5 numbers that are not practical are m = 3, 5, 7, 9, 10. Their abundancy indices sigma(m)/m are 1.333..., 1.2, 1.142..., 1.444..., 1.8. The record values occur at 3, 9 and 10.

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; seq = {}; rm = 1; Do[fct = FactorInteger[n]; r = Times@@((First/@fct^ (1+Last/@ fct)-1)/(First/@fct-1))/n; If[r > rm && !pracQ[fct], rm = r; AppendTo[seq, n]], {n, 3, 10^5}]; seq

Crossrefs

Cf. A000203 (sigma), A004394, A005153, A237287, A330244, A335029.

Sequence in context: A031115 A004669 A173242 * A335029 A228450 A121057

Adjacent sequences: A335027 A335028 A335029 * A335031 A335032 A335033

Keyword

nonn,more

Author

Amiram Eldar, May 20 2020

Status

approved