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

1, 2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 660, 720, 840, 960, 1008, 1080, 1200, 1260, 1440, 1560, 1680, 1980, 2100, 2160, 2340, 2400, 2520, 2880, 3120, 3240

Offset

1,2

Comments

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

Links

Don Reble, Table of n, a(n) for n = 1..6524 (first 372 terms from T. D. Noe, terms 373 to 1000 from Donovan Johnson, terms 1001 to 2750 from Robert G. Wilson v)

Example

From William A. Tedeschi, Aug 19 2010: (Start)
-- 12: 1+2+3+4+6 = 16
13: 1 = 1
14: 1+2+7 = 10
15: 1+3+5 = 9
16: 1+2+4+8 = 15
17: 1 = 1
-- 18: 1+2+3+6+9 = 21
As 12 had the previous (earliest) highest, it is a term; then since 18 has the new highest, it is a term. (End)
Table of initial values of n, a(n), A034091(n) = f(a(n)), where f(k) = sigma(k)-k = A001065(k):
1, 1, 0
2, 2, 1
3, 4, 3
4, 6, 6
5, 8, 7
6, 10, 8
7, 12, 16
8, 18, 21
9, 20, 22
10, 24, 36
11, 30, 42
12, 36, 55
13, 48, 76
14, 60, 108
15, 72, 123
16, 84, 140
17, 90, 144
18, 96, 156
19, 108, 172
20, 120, 240

Mathematica

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

Prog

(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

Crossrefs

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

Supersequence of A002093.

Cf. A001065, A034091, A034287, A034288.

Sequence in context: A100180 A258137 A101814 * A146344 A162763 A113242

Adjacent sequences: A034087 A034088 A034089 * A034091 A034092 A034093

Keyword

nonn,nice

Author

J. Lowell

Extensions

More terms from Erich Friedman

Status

approved