A139228 First differences of perfect numbers A000396.

22, 468, 7632, 33542208, 8556318720, 128848822272, 2305842870701260800, 2658455991569831742348849607813890048, 191561942608236104636337386514471893476304705594327040

Offset: 1

Author

Omar E. Pol, Apr 18 2008

Keywords

• nonn

Examples

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

Links

• Paolo Xausa, Table of n, a(n) for n = 1..14
• Philippe Ellia, On the distance between perfect numbers, arXiv:1210.0450 [math.NT], 2012.
• Florian Luca, Problem 10711, Amer. Math. Monthly, Vo. 106, No. 2 (1999) p. 166; Can Two Consecutive Numbers Both Be Perfect?, solution by Francis B. Coghlan, ibid., Vol. 108, No. 1 (2001), pp. 80-81.
• Florian Luca and Carl Pomerance, On the radical of a perfect number, New York Journal of Math., Vol. 16 (2010), pp. 23-30; alternative link.
• Florian Luca and Herman te Riele, phi and sigma: from Euler to Erdős, Nieuw Archief voor Wiskunde, Vol. 12, No. 5 (2011), pp. 31-36.

Crossrefs

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

Programs

Mathematica

Differences[Select[Range[10000], DivisorSigma[1, #] == 2# &]] (* Alonso del Arte, Mar 05 2020 *)
Differences[PerfectNumber[Range[12]]] (* Paolo Xausa, Oct 20 2023 *)

Formula

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

From Amiram Eldar, May 07 2021: (Start)

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

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

Extensions

More terms from Omar E. Pol, Oct 02 2012