A067803 -id:A067803 - cp's OEIS frontend

A223136 Numbers n such that sigma(n+1) - sigma(n) = k*n for some integer k, where sigma(n) = A000203 (sum of divisors of n).

1, 3, 7, 14, 31, 127, 206, 532, 954, 957, 1334, 1364, 1634, 2685, 2974, 4364, 8191, 14841, 18873, 19358, 20145, 24957, 33998, 36566, 42818, 56564, 64665, 74918, 79826, 79833, 84134, 92685, 104944, 109214, 111506, 116937, 122073, 131071, 138237, 147454, 161001

Offset: 1

Jaroslav Krizek, May 01 2013

nonn

Supersequence of A000668 for k=1 (Mersenne primes), A067803 for k=-1 (numbers n such that sigma(n) - sigma(n+1) = n) and A002961 for k=0 (numbers n such that n and n+1 have same sum of divisors). For number 1 is k=2.

Corresponding values of integers k: 2, 1, 1, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,....

			Number 532 is in sequence because sigma(533) - sigma(532) = 588 - 1120 = -532 = (-1) * 532; k = -1.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000668, A067803, A002961, A000203.

Select[Range[10000], IntegerQ[(DivisorSigma[1, # + 1] - DivisorSigma[1, #])/#] &] (* T. D. Noe, May 02 2013 *)

Extended by T. D. Noe, May 02 2013

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

Original entry on oeis.org

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

Jaroslav Krizek, May 01 2013

nonn

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

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

Cf. A000203, A000668, A067803.

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 *)

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

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

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

cp's OEIS Frontend

A223136 Numbers n such that sigma(n+1) - sigma(n) = k*n for some integer k, where sigma(n) = A000203 (sum of divisors of n).

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