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A194472
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Erdős-Nicolas numbers.
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8
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24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800, 91963648, 211891200, 1931236608, 2013143040, 4428914688, 10200236032, 214204956672
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OFFSET
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1,1
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COMMENTS
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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
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
Named after the Hungarian mathematician Paul Erdős (1913-1996) and the French mathematician Jean-Louis Nicolas. - Amiram Eldar, Jun 23 2021
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REFERENCES
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Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 141.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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])))
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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