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%I #111 Feb 18 2022 19:28:54
%S 2,4,16,64,4096,65536,262144,1073741824,1152921504606846976
%N Superperfect numbers: numbers k such that sigma(sigma(k)) = 2*k where sigma is the sum-of-divisors function (A000203).
%C Let sigma_m(n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives (2,2)-perfect numbers.
%C Even values of these are 2^(p-1) where 2^p-1 is a Mersenne prime (A000043 and A000668). No odd superperfect numbers are known. Hunsucker and Pomerance checked that there are no odd ones below 7 * 10^24. - _Jud McCranie_, Jun 01 2000
%C The number of divisors of a(n) is equal to A000043(n), if there are no odd superperfect numbers. - _Omar E. Pol_, Feb 29 2008
%C The sum of divisors of a(n) is the n-th Mersenne prime A000668(n), provided that there are no odd superperfect numbers. - _Omar E. Pol_, Mar 11 2008
%C Largest proper divisor of A072868(n) if there are no odd superperfect numbers. - _Omar E. Pol_, Apr 25 2008
%C This sequence is a divisibility sequence if there are no odd superperfect numbers. - _Charles R Greathouse IV_, Mar 14 2012
%C For n>1, sigma(sigma(a(n))) + phi(phi(a(n))) = (9/4)*a(n). - _Farideh Firoozbakht_, Mar 02 2015
%C The term "super perfect number" was coined by Suryanarayana (1969). He and Kanold (1969) gave the general form of even superperfect numbers. - _Amiram Eldar_, Mar 08 2021
%D Dieter Bode, Über eine Verallgemeinerung der vollkommenen Zahlen, Dissertation, Braunschweig, 1971.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B9, pp. 99-100.
%D József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, pp. 110-111.
%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, pp. 38-42.
%H P. Bundschuh, <a href="https://eudml.org/doc/140933">Aufgabe 601</a>, Elem. Math., Vol. 24 (1969), p. 69; <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1969%3A24%3A%3A7#71">alternative link</a>.
%H G. L. Cohen and H. J. J. te Riele, <a href="http://projecteuclid.org/euclid.em/1047565640">Iterating the sum-of-divisors function</a>, Experimental Mathematics, 5 (1996), pp. 93-100.
%H G. G. Dandapat, J. L. Hunsucker, and Carl Pomerance, <a href="http://projecteuclid.org/euclid.pjm/1102905990">Some new results on odd perfect numbers</a>, Pacific J. Math. Volume 57, Number 2 (1975), 359-364.
%H A. Hoque and H. Kalita, <a href="http://www.naturalspublishing.com/files/published/1r9c4i46d2gg27.pdf">Generalized perfect numbers connected with arithmetic functions</a>, Math. Sci. Lett. 3, No. 3, 249-253 (2014).
%H J. L. Hunsucker and Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/super.pdf">There are no odd superperfect number less than 7*10^24</a>, Indian J. Math., Vol. 17 (1975), pp. 107-120.
%H H.-J. Kanold, <a href="https://eudml.org/doc/140929">Über "Super perfect numbers"</a>, Elem. Math., Vol. 24 (1969), pp. 61-62; <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1969%3A24%3A%3A7#67">alternative link</a>.
%H Graham Lord, <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1975%3A30%3A%3A7#90">Even Perfect and Superperfect Numbers</a>, Elem. Math., Vol. 30 (1975), pp. 87-88.
%H H. G. Niederreiter, <a href="https://eudml.org/doc/140973">Solution of Aufgabe 601</a>, Elem. Math., Vol. 25 (1970), pp. 66-67; <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1970%3A25%3A%3A7#69">alternative link</a>.
%H Paul Shubhankar, <a href="https://www.erpublication.org/published_paper/IJETR011954.pdf">Ten Problems of Number Theory</a>, International Journal of Engineering and Technical Research (IJETR), ISSN: 2321-0869, Volume-1, Issue-9, November 2013.
%H D. Suryanarayana, <a href="https://eudml.org/doc/140912">Super perfect numbers</a>, Elem. Math., Vol. 24 (1969), pp. 16-17; <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1969%3A24%3A%3A7#22">alternative link</a>.
%H D. Suryanarayana, <a href="https://eudml.org/doc/141127">There is no superperfect number of the form p^(2*alpha)</a>, Elem. Math., Vol. 28 (1973), pp. 148-150; <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1973%3A28%3A%3A31#155">alternative link</a>.
%H László Tóth, <a href="http://macs.elte.hu/downloads/abstracts/MaCS_abs_Toth.pdf">The alternating sum-of-divisors function</a>, 9th Joint Conf. on Math. and Comp. Sci., February 9-12, 2012, Siófok, Hungary.
%H László Tóth, <a href="http://arxiv.org/abs/1111.4842">A survey of the alternating sum-of-divisors function</a>, arXiv:1111.4842 [math.NT], 2011-2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperperfectNumber.html">Superperfect Number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Superperfect_number">Superperfect number</a>.
%H Tomohiro Yamada, <a href="https://jtnb.centre-mersenne.org/item/JTNB_2020__32_1_259_0/">On finiteness of odd superperfect numbers</a>, Journal de Théorie des Nombres de Bordeaux, Vol. 32, No. 1 (2020), pp. 259-274.
%F a(n) = (1 + A000668(n))/2, if there are no odd superperfect numbers. - _Omar E. Pol_, Mar 11 2008
%F Also, if there are no odd superperfect numbers then a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - _Omar E. Pol_, Apr 25 2008
%F a(n) = 2^A090748(n), if there are no odd superperfect numbers. - _Ivan N. Ianakiev_, Sep 04 2013
%e sigma(sigma(4))=2*4, so 4 is in the sequence.
%t sigma = DivisorSigma[1, #]&;
%t For[n = 2, True, n++, If[sigma[sigma[n]] == 2 n, Print[n]]] (* _Jean-François Alcover_, Sep 11 2018 *)
%o (PARI) is(n)=sigma(sigma(n))==2*n \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Python)
%o from itertools import count, islice
%o def A019279_gen(): # generator of terms
%o return (n for n in count(1) if divisor_sigma(divisor_sigma(n)) == 2*n)
%o A019279_list = list(islice(A019279_gen(),6)) # _Chai Wah Wu_, Feb 18 2022
%Y Cf. A019280, A000203, A000396, A000668, A000043, A034897, A061652, A032742, A072868.
%K nonn,more,nice
%O 1,1
%A _N. J. A. Sloane_
%E a(8)-a(9) from _Jud McCranie_, Jun 01 2000
%E Corrected by _Michel Marcus_, Oct 28 2017
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