A125639 Doubly abundant numbers - numbers k such that k and s(k) are abundant, where s() is A001065.

24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 114, 120, 126, 132, 138, 140, 150, 168, 174, 176, 180, 186, 198, 204, 210, 216, 222, 224, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282, 294, 306, 308, 312, 318, 330, 340, 342, 348, 354, 360, 364, 366, 378, 380

Offset: 1

Author

Gabriel Cunningham (gabriel.cunningham(AT)gmail.com), Nov 28 2006

Keywords

• nonn

Comments

Unlike abundant numbers, not all multiples of doubly abundant numbers are doubly abundant; for instance, 48 is not doubly abundant. There are infinitely many doubly abundant numbers; for instance, all numbers of the form 24*25^k are doubly abundant. Such a number is abundant, being a multiple of an abundant number and s(24*25^k) = s(24)*s(25^k) + 24*s(25^k) + 25^k*s(24), which is a multiple of s(24) = 36.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005101, A125640, A001065.

Programs

Haskell

a125639 n = a125639_list !! (n-1)
a125639_list = filter f [1..] where
   f x = sx > x && a001065 sx > sx where sx = a001065 x
-- Reinhard Zumkeller, Oct 31 2015

Mathematica

s[n_] := DivisorSigma[1, n] - n; q[n_] := Module[{s1 = s[n]}, s1 > n && s[s1] > s1]; Select[Range[400], q] (* Amiram Eldar, Mar 11 2024 *)

PARI

is(n)=my(s=sigma(n)); s>2*n && sigma(s-n,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017