A194472 Erdős-Nicolas numbers.

24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800, 91963648, 211891200, 1931236608, 2013143040, 4428914688, 10200236032, 214204956672

Offset: 1

Author

Alonso del Arte, Aug 24 2011

Keywords

• nonn
• more

Comments

Abundant numbers m such that the sum of the first k divisors is equal to m for some k, thus this is a subsequence of A064510. k has to be less than tau(m) - 1 for this sequence, whereas in A064510 k = tau(m) - 1 is allowed (and thus perfect numbers are in that sequence).

a(17) > 5*10^11. 104828758917120, 916858574438400, 967609154764800, 93076753068441600, 215131015678525440 and 1371332329173024768 are also terms. - Donovan Johnson, Dec 26 2012

a(17) > 10^12. - Giovanni Resta, Apr 15 2017

Equivalently, numbers whose abundancy equals 1 + the sum of the reciprocals of its first k divisors for some k > 1. - Charlie Neder, Feb 08 2019

96892692739248881664, 41407449045801454927872, 101616496263816777695232, 1346571992706422996646631651147776, 3304572752464376776401640967110656 are also terms. - Michel Marcus, Feb 09 2019

All known terms of A141643 (abundancy 5/2) are terms. - Michel Marcus, Feb 11 2019

Named after the Hungarian mathematician Paul Erdős (1913-1996) and the French mathematician Jean-Louis Nicolas. - Amiram Eldar, Jun 23 2021

Are all terms in this sequence even? - Jenaro Tomaszewski, May 07 2023

Examples

			The divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24, hence 24 is in the list.
The divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The first seven of these add up to 36, but the first eight add up to 52, therefore 48 is not on the list.
		

References

• Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 141.

Links

• P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math. France, Vol. 103, No. 1 (1975), pp. 65-90.
• Wikipedia, Erdős-Nicolas number.

Crossrefs

Cf. A005835, A000396, A064510, A141643.

Programs

Mathematica

subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[Divisors[n], -2]]; erdNickNums = {}; Do[If[selDivs[n] == 0, AppendTo[erdNickNums, n]], {n, 2, 10^5}]; erdNickNums (* Based on the program by Bobby R. Treat and Robert G. Wilson v for A064510 *)

PARI

isok(n) = {if (sigma(n) <= 2*n, return (0)); my(d = divisors(n), s = 0); for (k=1, #d-2, s += d[k]; if (s == n, return (1)); if (s > n, break);); return (0);} \\ Michel Marcus, Feb 09 2019

Python

from itertools import accumulate, count, islice
from sympy import divisors
def A194472_gen(startvalue=1): # generator of terms >= startvalue
    return (n for n in count(max(startvalue,1)) if any(s == n for s in accumulate(divisors(n)[:-2])))
A194472_list = list(islice(A194472_gen(),5)) # Chai Wah Wu, Feb 18 2022

Extensions

More terms from M. F. Hasler, Aug 24 2011