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A034090 Numbers k whose sum of proper divisors (A001065(k)) exceeds that of all smaller numbers.

%I A034090 #52 Feb 11 2023 20:23:29
%S A034090 1,2,4,6,8,10,12,18,20,24,30,36,48,60,72,84,90,96,108,120,144,168,180,
%T A034090 216,240,288,300,336,360,420,480,504,540,600,660,720,840,960,1008,
%U A034090 1080,1200,1260,1440,1560,1680,1980,2100,2160,2340,2400,2520,2880,3120,3240
%N A034090 Numbers k whose sum of proper divisors (A001065(k)) exceeds that of all smaller numbers.
%C A034090 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 A034090 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 A034090 From _William A. Tedeschi_, Aug 19 2010: (Start)
%e A034090 -- 12: 1+2+3+4+6 = 16
%e A034090 13: 1 = 1
%e A034090 14: 1+2+7 = 10
%e A034090 15: 1+3+5 = 9
%e A034090 16: 1+2+4+8 = 15
%e A034090 17: 1 = 1
%e A034090 -- 18: 1+2+3+6+9 = 21
%e A034090 As 12 had the previous (earliest) highest, it is a term; then since 18 has the new highest, it is a term. (End)
%e A034090 Table of initial values of n, a(n), A034091(n) = f(a(n)), where f(k) = sigma(k)-k = A001065(k):
%e A034090 1, 1, 0
%e A034090 2, 2, 1
%e A034090 3, 4, 3
%e A034090 4, 6, 6
%e A034090 5, 8, 7
%e A034090 6, 10, 8
%e A034090 7, 12, 16
%e A034090 8, 18, 21
%e A034090 9, 20, 22
%e A034090 10, 24, 36
%e A034090 11, 30, 42
%e A034090 12, 36, 55
%e A034090 13, 48, 76
%e A034090 14, 60, 108
%e A034090 15, 72, 123
%e A034090 16, 84, 140
%e A034090 17, 90, 144
%e A034090 18, 96, 156
%e A034090 19, 108, 172
%e A034090 20, 120, 240
%t A034090 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 A034090 DeleteDuplicates[Table[{n,DivisorSigma[1,n]-n},{n,5000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* _Harvey P. Dale_, Jan 15 2023 *)
%o A034090 (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 A034090 This sequence and A034091 together give the record high points in A001065.
%Y A034090 Supersequence of A002093.
%Y A034090 Cf. A001065, A034091, A034287, A034288.
%K A034090 nonn,nice
%O A034090 1,2
%A A034090 _J. Lowell_
%E A034090 More terms from _Erich Friedman_