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%I #71 Aug 05 2023 23:26:57
%S 24,2016,8190,42336,45864,392448,714240,1571328,61900800,91963648,
%T 211891200,1931236608,2013143040,4428914688,10200236032,214204956672
%N Erdős-Nicolas numbers.
%C 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).
%C a(17) > 5*10^11. 104828758917120, 916858574438400, 967609154764800, 93076753068441600, 215131015678525440 and 1371332329173024768 are also terms. - _Donovan Johnson_, Dec 26 2012
%C a(17) > 10^12. - _Giovanni Resta_, Apr 15 2017
%C 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
%C 96892692739248881664, 41407449045801454927872, 101616496263816777695232, 1346571992706422996646631651147776, 3304572752464376776401640967110656 are also terms. - _Michel Marcus_, Feb 09 2019
%C All known terms of A141643 (abundancy 5/2) are terms. - _Michel Marcus_, Feb 11 2019
%C Named after the Hungarian mathematician Paul Erdős (1913-1996) and the French mathematician Jean-Louis Nicolas. - _Amiram Eldar_, Jun 23 2021
%C Are all terms in this sequence even? - _Jenaro Tomaszewski_, May 07 2023
%D Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 141.
%H P. Erdős and J.-L. Nicolas, <a href="http://www.renyi.hu/~p_erdos/1975-37.pdf">Répartition des nombres superabondants</a>, Bull. Soc. Math. France, Vol. 103, No. 1 (1975), pp. 65-90.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Nicolas_number">Erdős-Nicolas number</a>.
%e 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.
%e 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.
%t 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 *)
%o (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
%o (Python)
%o from itertools import accumulate, count, islice
%o from sympy import divisors
%o def A194472_gen(startvalue=1): # generator of terms >= startvalue
%o return (n for n in count(max(startvalue,1)) if any(s == n for s in accumulate(divisors(n)[:-2])))
%o A194472_list = list(islice(A194472_gen(),5)) # _Chai Wah Wu_, Feb 18 2022
%Y Cf. A005835, A000396, A064510, A141643.
%K nonn,more
%O 1,1
%A _Alonso del Arte_, Aug 24 2011
%E More terms from _M. F. Hasler_, Aug 24 2011
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