A318168 Reverse Erdős-Nicolas numbers: abundant numbers m such that the sum of the last k proper divisors of m is equal to m for some k.

18, 42, 54, 66, 78, 102, 114, 126, 138, 162, 174, 186, 196, 198, 222, 234, 246, 258, 282, 294, 306, 318, 342, 354, 366, 378, 402, 414, 426, 438, 462, 474, 486, 498, 522, 534, 546, 558, 582, 594, 606, 618, 642, 654, 666, 678, 702, 714, 726, 738, 762, 774, 786

Offset: 1

Author

Amiram Eldar, Aug 20 2018

Keywords

• nonn

Comments

Apparently most of the terms are sum of their 3 largest proper divisors and are included in A074837. Terms that are not there are 196, 812, 868, 1036, 1148, 1204, 1316, 1372, 1484, 1652, 1708, 1876, 1998, 2044, ...

The possible values of k seem to be rather sparse. Up to 2*10^10, such values are: 3 (minimal m = 18), 5 (196), 9 (15376), 13 (1032256), 15 (34155), 16 (20482), 17 (33345), 19 (8415), 21 (407715), 23 (1273725), 26 (89245784), 32 (479198624), 36 (125226568), 40 (12499150), 45 (5905148248), 46 (1375270384), 68 (13968326788), and 91 (159030135). - Giovanni Resta, Aug 21 2018

If 2^p - 1 is prime then ((2^p - 1)^n)*2^(p-1) is in the sequence for n > 1. - Davide Rotondo, Oct 02 2021

From Mauro Fiorentini, Jan 08 2024: (Start)

More generally, if n is an even perfect number, with odd prime factor p, all prime factors of m are greater than n, k >= 0 and p^k*m > 1, n*p^k*m is in the sequence.

Also, if n is in the sequence and all prime factors of m are greater than n, n*m is in the sequence (note that n is not necessarily a multiple of an even perfect number).

It follows that there are infinitely many odd terms in the sequence, that the asymptotic density of the sequence is greater than 0.073482 and that the difference between consecutive terms is at most 24. (End)

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 5, 68, 737, 7352, 73704, 737142, 7370307, 73699222, 737011233, 7370145824, ... . Apparently, the asymptotic density of this sequence exists and equals 0.073701... . - Amiram Eldar, Apr 18 2024

Examples

			196 is in the sequence since its proper divisors are 1, 2, 4, 7, 14, 28, 49, 98, and 7 + 14 + 28 + 49 + 98 = 196.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000043, A000668, A074837, A194472.

Programs

Mathematica

subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Reverse[Rest[Most[Divisors[n]]]]]; s={}; Do[If[selDivs[n] == 0, AppendTo[s, n]], {n, 2, 1000}]; s

PARI

isok(n) = {my(d = Vecrev(divisors(n))); if (vecsum(d) > 2*n, my(s = 0); for (i=2, #d, s += d[i]; if (s == n, return(n)););); return (0);} \\ Michel Marcus, Aug 21 2018