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%I #20 Dec 21 2023 17:00:26
%S 2,3,4,7,8,20,26,31,127,532,954,2186,2524,8191,104944,131071,524287,
%T 918080,1594322,10368512,26100416,2147483647,24708617408,25316030960,
%U 35053995440,45883878740
%N Numbers n such that sigma(n+1) - sigma(n) divides n.
%C Supersequence of A000668 (Mersenne primes) and A067803 (numbers n such that sigma(n) - sigma(n+1) = n).
%C Corresponding integers k such that sigma(n+1) - sigma(n) = n/k: 2, 1, -4, 1, -4, -2, -13, 1, 1, -1, -1, -1093, -2, 1, ....
%C a(27) > 10^11. - _Donovan Johnson_, May 04 2013
%C a(27) > 10^13. - _Giovanni Resta_, Aug 01 2013
%e Number 2186 is in sequence because sigma(2187) - sigma(2186) = 3280 - 3282 = -2 which divides 2186.
%t Select[Range[1000000], DivisorSigma[1, # + 1] - DivisorSigma[1, #] != 0 && IntegerQ[#/(DivisorSigma[1, # + 1] - DivisorSigma[1, #])] &] (* _T. D. Noe_, May 02 2013 *)
%t Select[Partition[Table[{n,DivisorSigma[1,n]},{n,16*10^5}],2,1],Divisible[#[[1,1]],#[[1,2]]-#[[2,2]]]&][[All,1,1]]//Quiet (* _Harvey P. Dale_, Jun 27 2020 *)
%t Position[MapIndexed[Divisible[#2[[1]], #] &, Subtract @@@ Partition[DivisorSigma[1, Range[10^6]], 2, 1]], True] // Flatten // Quiet (* _Eric W. Weisstein_, Dec 21 2023 *)
%o (PARI) is(n)=my(m=sigma(n+1)-sigma(n)); m && n%m==0 \\ _Charles R Greathouse IV_, May 02 2013
%Y Cf. A000203, A000668, A067803.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, May 01 2013
%E a(15)-a(21) from _Charles R Greathouse IV_, May 02 2013
%E a(22)-a(26) from _Donovan Johnson_, May 04 2013
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