A225211 Numbers n such that sigma(n+1) - sigma(n) divides n.

2, 3, 4, 7, 8, 20, 26, 31, 127, 532, 954, 2186, 2524, 8191, 104944, 131071, 524287, 918080, 1594322, 10368512, 26100416, 2147483647, 24708617408, 25316030960, 35053995440, 45883878740

Offset: 1

Author

Jaroslav Krizek, May 01 2013

Keywords

• nonn

Comments

Supersequence of A000668 (Mersenne primes) and A067803 (numbers n such that sigma(n) - sigma(n+1) = n).

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, ....

a(27) > 10^11. - Donovan Johnson, May 04 2013

a(27) > 10^13. - Giovanni Resta, Aug 01 2013

Examples

			Number 2186 is in sequence because sigma(2187) - sigma(2186) = 3280 - 3282 = -2 which divides 2186.
		

Crossrefs

Cf. A000203, A000668, A067803.

Programs

Mathematica

Select[Range[1000000], DivisorSigma[1, # + 1] - DivisorSigma[1, #] != 0 && IntegerQ[#/(DivisorSigma[1, # + 1] - DivisorSigma[1, #])] &] (* T. D. Noe, May 02 2013 *)
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 *)
Position[MapIndexed[Divisible[#2[[1]], #] &, Subtract @@@ Partition[DivisorSigma[1, Range[10^6]], 2, 1]], True] // Flatten // Quiet (* Eric W. Weisstein, Dec 21 2023 *)

PARI

is(n)=my(m=sigma(n+1)-sigma(n)); m && n%m==0 \\ Charles R Greathouse IV, May 02 2013

Extensions

a(15)-a(21) from Charles R Greathouse IV, May 02 2013

a(22)-a(26) from Donovan Johnson, May 04 2013