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A139228 First differences of perfect numbers A000396. 12

%I #31 Oct 20 2023 08:40:37

%S 22,468,7632,33542208,8556318720,128848822272,2305842870701260800,

%T 2658455991569831742348849607813890048,

%U 191561942608236104636337386514471893476304705594327040

%N First differences of perfect numbers A000396.

%H Paolo Xausa, <a href="/A139228/b139228.txt">Table of n, a(n) for n = 1..14</a>

%H Philippe Ellia, <a href="http://arxiv.org/abs/1210.0450">On the distance between perfect numbers</a>, arXiv:1210.0450 [math.NT], 2012.

%H Florian Luca, <a href="https://www.jstor.org/stable/2589055">Problem 10711</a>, Amer. Math. Monthly, Vo. 106, No. 2 (1999) p. 166; <a href="https://www.jstor.org/stable/2695692">Can Two Consecutive Numbers Both Be Perfect?</a>, solution by Francis B. Coghlan, ibid., Vol. 108, No. 1 (2001), pp. 80-81.

%H Florian Luca and Carl Pomerance, <a href="http://nyjm.albany.edu/j/2010/16-3.html">On the radical of a perfect number</a>, New York Journal of Math., Vol. 16 (2010), pp. 23-30; <a href="http://www.math.dartmouth.edu/~carlp/LucaPomeranceNYJMstyle.pdf">alternative link</a>.

%H Florian Luca and Herman te Riele, <a href="http://www.nieuwarchief.nl/serie5/pdf/naw5-2011-12-1-031.pdf">phi and sigma: from Euler to Erdős</a>, Nieuw Archief voor Wiskunde, Vol. 12, No. 5 (2011), pp. 31-36.

%F a(n) = A000396(n+1) - A000396(n).

%F From _Amiram Eldar_, May 07 2021: (Start)

%F a(n) > 1 (Luca, 1999).

%F a(n) > 4 (Luca and te Riele, 2011). (End)

%e a(1) = 22 because 6 and 28 are the first two perfect numbers, and their difference is 28 - 6 = 22.

%t Differences[Select[Range[10000], DivisorSigma[1, #] == 2# &]] (* _Alonso del Arte_, Mar 05 2020 *)

%t Differences[PerfectNumber[Range[12]]] (* _Paolo Xausa_, Oct 20 2023 *)

%Y Cf. A000396, A139229, A139230, A139231, A139232, A139233, A139234, A139235, A139236, A139237.

%K nonn

%O 1,1

%A _Omar E. Pol_, Apr 18 2008

%E More terms from _Omar E. Pol_, Oct 02 2012

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)