A083347 Numbers k such that Sum(e/p: k=Product(p^e)) < 1.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

Author

Reinhard Zumkeller, Apr 25 2003

Keywords

• nonn

Comments

Numbers k whose arithmetic derivative (A003415) k' < k. - T. D. Noe, Apr 24 2011

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003415, A072873, A051674, A083348.

Cf. A168036.

Programs

Haskell

a083347 n = a083347_list !! (n-1)
a083347_list = filter ((< 0) . a168036) [1..]
-- Reinhard Zumkeller, May 22 2015, May 10 2011

Mathematica

Select[Range@ 102, If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger@ Abs@ #]] < # &] (* Michael De Vlieger, Feb 02 2019 *)

Formula

A083345(a(n)) < A083346(a(n));

A083345(A051674(n)) = A083346(A051674(n));

A083345(A083348(n)) > A083346(A083348(n)).

A168036(a(n)) < 0. - Reinhard Zumkeller, May 22 2015