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Numbers n such that sigma(n+1) - sigma(n-1) = k*n for some integer k, where sigma(n) = A000203 (sum of divisors of n).
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%I #8 May 03 2013 20:56:41

%S 5,34,55,285,367,835,849,919,1241,1505,2911,2914,3305,4149,4188,6111,

%T 6903,7170,7913,9360,10251,10541,12566,15086,17273,17815,19005,19689,

%U 21411,21462,24882,25020,26610,28125,30593,30789,31485,38211,38983,39787,40311,45355

%N Numbers n such that sigma(n+1) - sigma(n-1) = k*n for some integer k, where sigma(n) = A000203 (sum of divisors of n).

%C Supersequence of A055574 for k=0 (n satisfying sigma(n+1) = sigma(n-1)). For number 5 is k=1. Are there other such number for k=1 or k=-1?

%C Corresponding values of integers k: 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,....

%H Donovan Johnson, <a href="/A223137/b223137.txt">Table of n, a(n) for n = 1..1000</a>

%e Number 5 is in sequence because sigma(6) - sigma(4) = 12 - 7 = 5 = 1 * 5; k=1.

%t Select[Range[100000], IntegerQ[(DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1])/#] &] (* _T. D. Noe_, May 02 2013 *)

%Y Cf. A055574, A000203.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, May 01 2013

%E Extended by _T. D. Noe_, May 02 2013