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%I M4297 #86 Apr 03 2023 10:36:09
%S 1,6,120,30240,14182439040,154345556085770649600,
%T 141310897947438348259849402738485523264343544818565120000
%N a(n) = first n-fold perfect (or n-multiperfect) number.
%C On the Riemann Hypothesis, a(n) > exp(exp(n / e^gamma)) for n > 3. Unconditionally, a(n) > exp(exp(0.9976n / e^gamma)) for n > 3, where the constant is related to A004394(1000000). - _Charles R Greathouse IV_, Sep 06 2012
%C Each of the terms 1, 6, 120, 30240 divides all larger terms <= a(8). See A227765, A227766, ..., A227769. - _Jonathan Sondow_, Jul 30 2013
%C Is a(n) < a(n+1)? - _Jeppe Stig Nielsen_, Jun 16 2015
%C Equivalently, a(n) is the smallest number k such that sigma(k)/k = n. - _Derek Orr_, Jun 19 2015
%C The number of divisors of these terms are: 1, 4, 16, 96, 1920, 110592, 1751777280, 63121588161085440. - _Michel Marcus_, Jun 20 2015
%C Given n, let S_n be the sequence of integers k that satisfy numerator(sigma(k)/k) = n. Then a(n) is a member of S_n. In fact a(n) = S_n(i), where the successive values of i are 1, 1, 2, 2, 4, 2, (23, 6, 31, 12, ...), where the terms in parentheses need to be confirmed. - _Michel Marcus_, Nov 22 2015
%C The first four terms are the only multiperfect numbers in A025487 among the 1600 initial terms of A007691. Can it be proved that these are the only ones among the whole A007691? See also A349747. - _Antti Karttunen_, Dec 04 2021
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
%D A. Brousseau, Number Theory Tables. Fibonacci Association, San Jose, CA, 1973, p. 138.
%D R. K. Guy, Unsolved Problems in Number Theory, B2.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007539/b007539.txt">Table of n, a(n) for n = 1..8</a>
%H Abiodun E. Adeyemi, <a href="https://arxiv.org/abs/1906.05798">A Study of @-numbers</a>, arXiv:1906.05798 [math.NT], 2019.
%H Jamie Bishop, Abigail Bozarth, Rebekah Kuss, and Benjamin Peet, <a href="https://arxiv.org/abs/2104.11366">The Abundancy Index and Feebly Amicable Numbers</a>, arXiv:2104.11366 [math.NT], 2021.
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Multiplyperfect">multiply perfect</a>
%H F. Firoozbakht and M. F. Hasler, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Hasler/hasler2.html">Variations on Euclid's formula for Perfect Numbers</a>, JIS 13 (2010) #10.3.1.
%H Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">The Multiply Perfect Numbers Page</a>
%H Fred Helenius, <a href="http://pw1.netcom.com/~fredh/index.html">Link to Glossary and Lists</a>
%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiperfect">Multiperfect and hemiperfect numbers</a>
%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy: Some Resources </a>
%t Table[k = 1; While[DivisorSigma[1, k]/k != n, k++]; k, {n, 4}] (* _Michael De Vlieger_, Jun 20 2015 *)
%o (PARI) a(n)=k=1;while((sigma(k)/k)!=n,k++);k
%o vector(4,n,a(n)) \\ _Derek Orr_, Jun 19 2015
%Y Cf. A000396, A005820, A007691, A025487, A027687, A046060, A046061, A072002, A227765, A227766, A227767, A227768, A227769, A349747.
%K nonn,nice,more
%O 1,2
%A _N. J. A. Sloane_
%E More terms sent by _Robert G. Wilson v_, Nov 30 2000
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