%I #52 Feb 11 2023 20:23:29

%S 1,2,4,6,8,10,12,18,20,24,30,36,48,60,72,84,90,96,108,120,144,168,180,

%T 216,240,288,300,336,360,420,480,504,540,600,660,720,840,960,1008,

%U 1080,1200,1260,1440,1560,1680,1980,2100,2160,2340,2400,2520,2880,3120,3240

%N Numbers k whose sum of proper divisors (A001065(k)) exceeds that of all smaller numbers.

%C The highly abundant numbers A002093 are a subsequence since if sigma(k) - k > sigma(m) - m for all m < n then sigma(k) > sigma(m). - _Charles R Greathouse IV_, Sep 13 2016

%H Don Reble, <a href="/A034090/b034090.txt">Table of n, a(n) for n = 1..6524</a> (first 372 terms from _T. D. Noe_, terms 373 to 1000 from _Donovan Johnson_, terms 1001 to 2750 from Robert G. Wilson v)

%e From _William A. Tedeschi_, Aug 19 2010: (Start)

%e -- 12: 1+2+3+4+6 = 16

%e 13: 1 = 1

%e 14: 1+2+7 = 10

%e 15: 1+3+5 = 9

%e 16: 1+2+4+8 = 15

%e 17: 1 = 1

%e -- 18: 1+2+3+6+9 = 21

%e As 12 had the previous (earliest) highest, it is a term; then since 18 has the new highest, it is a term. (End)

%e Table of initial values of n, a(n), A034091(n) = f(a(n)), where f(k) = sigma(k)-k = A001065(k):

%e 1, 1, 0

%e 2, 2, 1

%e 3, 4, 3

%e 4, 6, 6

%e 5, 8, 7

%e 6, 10, 8

%e 7, 12, 16

%e 8, 18, 21

%e 9, 20, 22

%e 10, 24, 36

%e 11, 30, 42

%e 12, 36, 55

%e 13, 48, 76

%e 14, 60, 108

%e 15, 72, 123

%e 16, 84, 140

%e 17, 90, 144

%e 18, 96, 156

%e 19, 108, 172

%e 20, 120, 240

%t A = {}; mx = -1; For[ k = 1, k < 10000, k++, t = DivisorSigma[1, k] - k; If[ t > mx, mx = t; AppendTo[A, k]]]; A (* slightly modified by _Robert G. Wilson v_, Aug 28 2022 *)

%t DeleteDuplicates[Table[{n,DivisorSigma[1,n]-n},{n,5000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* _Harvey P. Dale_, Jan 15 2023 *)

%o (PARI) r=0; for(n=1,1e6, t=sigma(n)-n; if(t>r, r=t; print1(n", "))) \\ _Charles R Greathouse IV_, Sep 13 2016

%Y This sequence and A034091 together give the record high points in A001065.

%Y Supersequence of A002093.

%Y Cf. A001065, A034091, A034287, A034288.

%K nonn,nice

%O 1,2

%A _J. Lowell_

%E More terms from _Erich Friedman_