A055708 -id:A055708 - cp's OEIS frontend

A191363 Numbers m such that sigma(m) = 2*m - 2.

3, 10, 136, 32896, 2147516416

Offset: 1

Luis H. Gallardo, May 31 2011

Let k be a nonnegative integer such that F(k) = 2^(2^k) + 1 is prime (a Fermat prime A019434), then m = (F(k)-1)*F(k)/2 appears in the sequence.
Conjecture: a(1)=3 is the only odd term of the sequence.
Conjecture: All terms of the sequence are of the above form derived from Fermat primes.
The sequence has 5 (known) terms in common with sequences A055708 (k-1 | sigma(k)) and A056006 (k | sigma(k)+2) since {a(n)} is a subsequence of both.
The first five terms of the sequence are respectively congruent to 3, 4, 4, 4, 4 modulo 6.
After a(5) there are no further terms < 8*10^9.
Up to m = 1312*10^8 there are no further terms in the class congruent to 4 modulo 6.
a(6) > 10^12. - Donovan Johnson, Dec 08 2011
a(6) > 10^13. - Giovanni Resta, Mar 29 2013
a(6) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
See A125246 for numbers with deficiency 4, i.e., sigma(m) = 2*m - 4, and A141548 for numbers with deficiency 6. - M. F. Hasler, Jun 29 2016 and Jul 17 2016
A term m of this sequence multiplied by a prime p not dividing it is abundant if and only if p < m-1. For each of a(2..5) there is such a prime near this limit (here: 7, 127, 30197, 2147483647) such that a(k)*p is a primitive weird number, cf. A002975. - M. F. Hasler, Jul 19 2016
Any term m of this sequence can be combined with any term j of A088831 to satisfy the property (sigma(m) + sigma(j))/(m+j) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. [Proof: If m = a(n) and j = A088831(k), then sigma(m) = 2m-2 and sigma(j) = 2j+2. Thus, sigma(m) + sigma(j) = (2m-2) + (2j+2) = 2m + 2j = 2(m+j), which implies that (sigma(m) + sigma(j))/(m+j) = 2(m+j)/(m+j) = 2.] - Timothy L. Tiffin, Sep 13 2016
At least the first five terms are a subsequence of A295296 and of A295298. - David A. Corneth, Antti Karttunen, Nov 26 2017
Conjectures: all terms are second hexagonal numbers (A014105). There are no terms with middle divisors. - Omar E. Pol, Oct 31 2018
The symmetric representation of sigma(m) of each of the 5 numbers in the sequence consists of 2 parts of width 1 that meet at the diagonal (subsequence of A246955). - Hartmut F. W. Hoft, Mar 04 2022
The first five terms coincide with the sum of two successive terms of A058891. The same is not true for a(6), if such exists. - Omar E. Pol, Mar 03 2023

			For n=1, a(1) = 3 since sigma(3) = 4 = 2*3 - 2.

Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, and Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018.

Cf. A000203, A002975, A056006, A055708, A088831 (abundance 2).
Cf. A033880, A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A125248 (deficiency 16).
Cf. A295296, A295298.
Cf. A014105, A237593, A246955.
Cf. A058891.

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -2]; // Vincenzo Librandi, Sep 15 2016

ok[n_] := DivisorSigma[1,n] == 2*n-2; Select[ Table[ 2^(2^k-1) * (2^(2^k)+1), {k, 0, 5}], ok] (* Jean-François Alcover, Sep 14 2011, after conjecture *)
Select[Range[10^6], DivisorSigma[1, #] == 2 # - 2 &] (* Michael De Vlieger, Sep 14 2016 *)

zp(a,b) = {my(c,c1,s); c = a; c1 = 2*c-2;
while(c

a(k)=(2^2^k+1)<<(2^k-1) \\ For k<6. - M. F. Hasler, Jul 27 2016

a(n) = (A019434(n)-1)*A019434(n)/2 for all terms known so far. - M. F. Hasler, Jun 29 2016

A056006 Numbers k such that k | sigma(k) + 2.

Original entry on oeis.org

1, 3, 10, 136, 32896, 2147516416

Offset: 1

Robert G. Wilson v, Jul 24 2000

n | sigma(n) gives the multi-perfect numbers A007691, n | sigma(n)+1 if n is a power of 2 (A000079).
This contains A191363 as subsequence, so for any Fermat prime F(k) = 2^2^k+1, the triangular number A000217(2^2^k)=(F(k)-1)*F(k)/2 is in this sequence. See also A055708 which is identical up to the first term. - M. F. Hasler, Oct 02 2014
a(7) > 10^13. - Giovanni Resta, Jul 13 2015
a(7) > 10^18. - Max Alekseyev, May 27 2025

Cf. A000203, A045768, A055708.

Do[If[Mod[DivisorSigma[1, n]+2, n]==0, Print[n]], {n, 1, 7*10^8}]

for(n=1,5e9,if((sigma(n)+2)%n==0,print1(n", "))) \\ Charles R Greathouse IV, Jun 01 2011

a(6) from Charles R Greathouse IV, Jun 01 2011

Edited by M. F. Hasler, Oct 02 2014

A088820 Numbers k with abundance radius of 8, i.e., abs(sigma(k)-2*k) = 8.

Original entry on oeis.org

22, 56, 130, 184, 368, 836, 1012, 2272, 11096, 17816, 18904, 33664, 45356, 70564, 77744, 85936, 91388, 100804, 128768, 254012, 388076, 391612, 527872, 1090912, 2087936, 2291936, 13174976, 17619844, 29465852, 35021696, 45335936, 120888092, 260378492, 381236216

Offset: 1

Labos Elemer, Oct 20 2003

nonn

Original definition: Abundance-radius=8, that is Abs[sigma[n]-2n]=8 (either +8 or -8). A045770 from 3rd term complemented by -8 cases.

			22 is in the sequence since sigma(22) = 1 + 2 + 11 + 22 = 36 = 2*22 - 8.
56 is in the sequence since sigma(56) = 1 + 2 + 4 + 7 + 8 + 14 + 28 + 56 = 120 = 2*56 + 8. - _Michael B. Porter_, Jul 20 2016

Amiram Eldar, Table of n, a(n) for n = 1..66 (terms below 10^18, from the b-files at A088833 and A125247)

Disjoint union of A088833 (abundance 8) and A125247 (deficiency 8).
Cf. A045770, A045768, A055708, A077374, A088007, A088008, A088010, A088011, A088012, A088818, A088819.
Cf. A000203 (sigma), A033880 (abundance), A005100 (deficient numbers).

[n: n in [1..2*10^7] | Abs(DivisorSigma(1, n) - 2*n) eq 8]; // Vincenzo Librandi, Jul 20 2016

Select[Range[1, 10^6], Abs[DivisorSigma[1, #] - 2 #] == 8 &] (* Vincenzo Librandi, Jul 20 2016 *)

is(n)=abs(sigma(n)-2*n)==8 \\ Use, e.g., select(is,[1..10^5]*2). - M. F. Hasler, Jul 19 2016

More terms from David Wasserman, Aug 18 2005
Edited by M. F. Hasler, Jul 19 2016
a(33)-a(34) from Amiram Eldar, Mar 11 2025

A298563 Numbers k such that k - 2 | sigma(k).

Original entry on oeis.org

1, 3, 5, 6, 14, 44, 110, 152, 884, 2144, 8384, 18632, 116624, 8394752, 15370304, 73995392, 536920064, 2147581952, 34360131584, 27034175140420610, 36028797421617152, 576460753914036224

Offset: 1

Zdenek Cervenka, Jan 21 2018

Similar to A055708.
Sequence includes every number of the form 2^(j-1)*(2^j+3) such that 2^j+3 is prime (i.e., j is a term in A057732); terms of this form are 5, 14, 44, 152, 2144, 8384, 8394752, 536920064, 2147581952, 34360131584, ... - Jon E. Schoenfield, Jan 22 2018
Superset of A125246. - Giovanni Resta, Jan 23 2018
Contains 2 times odd terms of A191363. Also, if m is a term of A056006 and q := (sigma(m) + 2)/m is coprime to m, them q*m is a term. - Max Alekseyev, May 25 2025

			For k=44, sigma(k)/(k-2) = sigma(44)/(44-2) = 84/42 = 2, so 44 belongs to the sequence;
for k=110, sigma(k)/(k-2) = sigma(110)/(110-2) = 216/108 = 2, so 110 is also a term.

Cf. A055708, A057732, A125246.

[n: n in [3..10^7]| DivisorSigma(1, n) mod (n-2) eq 0]; // Vincenzo Librandi, Jan 22 2018

Select[Range[10^6], Divisible[DivisorSigma[1, #], # - 2] &] (* Michael De Vlieger, Jan 21 2018 *)

isok(k) = (k!=2) && !(sigma(k) % (k-2)); \\ Michel Marcus, Jan 22 2018

a(17)-a(18) from Robert G. Wilson v, Jan 21 2018
a(19) from Giovanni Resta, Jan 23 2018
a(20)-a(22) from Max Alekseyev, May 27 2025

A386390 Numbers k such that k-1 | sigma+(k) where sigma+ is A107758.

Original entry on oeis.org

2, 6, 66, 225, 8646, 101025, 149497986, 20412000225

Offset: 1

Michel Marcus, Aug 20 2025

Cf. A107758, A055708.

a107758[n_]:=DivisorSum[n, DivisorSigma[1, #] &, CoprimeQ[n/#, #] &];Select[Range[2,10^6],Divisible[a107758[#],#-1]&] (* James C. McMahon, Aug 21 2025 *)

isok(k) = if (k>1, !(sumdiv(k, d, if(gcd(k/d, d) == 1, sigma(d))) % (k-1)));

a(8) from Vincenzo Librandi, Aug 21 2025

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A191363 Numbers m such that sigma(m) = 2*m - 2.

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A056006 Numbers k such that k | sigma(k) + 2.

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A088820 Numbers k with abundance radius of 8, i.e., abs(sigma(k)-2*k) = 8.

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A298563 Numbers k such that k - 2 | sigma(k).

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A386390 Numbers k such that k-1 | sigma+(k) where sigma+ is A107758.

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