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%I #9 Sep 15 2019 07:56:54
%S 4,5,6,9,10,12,18,30,32,42,54,56,60,72,101,102,108,129,138,144,150,
%T 172,176,180,192,198,204,216,220,228,240,252,270,282,312,348,384,420,
%U 432,462,522,544,570,600,618,642,648,660,792,810,822,828,858,882,900,1020
%N Numbers n such that sigma(n+1) - sigma(n-1) = n / k for some integer, where sigma(n) = A000203 (sum of divisors of n).
%C Supersequence of A014574 for k = n/2 (average of twin prime pairs).
%C Corresponding values of integers k: 2, 1, 3, 3, -10, 6, 9, 15, 2, 21, 3, 7, 30, 36, -101, 51, 54, -43, 69, 12, 75, -2, -22, 90, 96, 99, 17, -27, -5, 114, 120, 7, 135, 141, 156, 174, 2, 210, 216, 231, 261, -8, 285, 300, 309, 321, 9, 330, -18, 405, 411, 414, 429, 441, 75, 510, ... (supersequence of A040040).
%H Amiram Eldar, <a href="/A223138/b223138.txt">Table of n, a(n) for n = 1..10000</a>
%e Number 5 is in sequence because sigma(6) - sigma(4) = 12 - 7 = 5; k=1.
%t Select[Range[1000], DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1] != 0 && IntegerQ[#/(DivisorSigma[1, # + 1] - DivisorSigma[1, # - 1])] &] (* _T. D. Noe_, May 02 2013 *)
%Y Cf. A000203, A014574, A040040.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, May 01 2013