A058073 -id:A058073 - cp's OEIS frontend

A163193 Numbers k such that sigma(k) = 2*sigma(k+1).

12, 70, 88, 204, 220, 1750, 1888, 2958, 8142, 8632, 9114, 14664, 18414, 18762, 20118, 20712, 25194, 45520, 64206, 65652, 65964, 77814, 79338, 79824, 85096, 90804, 103410, 103644, 117822, 158946, 163938, 176364, 185776, 186612, 194416, 202656, 203898, 245632

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 22 2009

nonn

The cases sigma(k) = 3*sigma(k+1) are rarer: k=180, 12000, 30996, 47940, ... [R. J. Mathar, Jul 25 2009]

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

Cf. A067081, A002961, A058073.

f[n_]:=DivisorSigma[1,n]; lst={};Do[If[f[n]==f[n+1]*2,AppendTo[lst,n]], {n,9!}];lst

{k: A000203(k) = 2*A000203(k+1)}.

Edited by R. J. Mathar, Jul 25 2009

A132585 Numbers k such that sigma(k)-k-1 divides sigma(k+1)-k-2, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

Original entry on oeis.org

25, 49, 799, 899, 32399, 292681, 1492995736325809

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

The banal case of ratio equal to zero is excluded. In fact if k+1 is a prime than sigma(k+1)-k-2=0. Therefore the ratio with sigma(k)-k-1 is equal to zero. Is this sequence finite?
a(7) <= 1492995736325809. [From Donovan Johnson, Aug 31 2008]
a(7) > 10^13. - Giovanni Resta, Jul 11 2013
No other terms < 2.7*10^15. - Jud McCranie, Jul 26 2025

			k=25 -> sigma(25)= 1+5+25 -> sigma(k)-k-1=5
k+1=26 -> sigma(26)= 1+2+13+26 -> sigma(k+1)-k-2=2+13=15
15/5 = 3 (integer > 0)

Cf. A000203, A002961, A058072, A058073, A132586.

with(numtheory); P:=proc(n) local a,i; for i from 1 by 1 to n do if sigma(i)-i-1>0 then a:=(sigma(i+1)-i-2)/(sigma(i)-i-1); if a>0 and trunc(a)=a then print(i); fi; fi; od; end: P(100000);

a(6) from Donovan Johnson, Aug 31 2008

a(7) by Jud McCranie, Jul 26 2025

A132586 Numbers k such that sigma(k+1)-k-2 divides sigma(k)-k-1, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

Original entry on oeis.org

8, 24, 8925, 32445, 118540859325

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

The banal case of ratio equal to zero is excluded. In fact if k is a prime than sigma(k)-k-1=0. Therefore the ratio with sigma(k+1)-k-2 is equal to zero. Is this sequence finite?
a(6), if it exists, is larger than 10^13. - Giovanni Resta, Jul 13 2015
No more terms < 2.7*10^15. - Jud McCranie, Jul 27 2025

			n=8 -> sigma(8)=1+2+4+8 -> sigma(n)-n-1=2+4=6.
n+1=9 -> sigma(9)=1+3+9 -> sigma(n+1)-n-2=3.
6/3 = 2 (integer >0)

Cf. A002961, A058072, A058073, A132585.

with(numtheory); P:=proc(n) local a,i; for i from 1 by 1 to n do if sigma(i+1)-i-2>0 then a:=(sigma(i)-i-1)/(sigma(i+1)-i-2); if a>0 and trunc(a)=a then print(i); fi; fi; od; end: P(100000);

a(5) from Donovan Johnson, Aug 31 2008

A217791 Numbers k such that sigma(k) = 3*sigma(k+1).

Original entry on oeis.org

180, 12000, 30996, 47940, 66780, 102816, 128040, 234300, 494088, 712272, 1133088, 1408212, 1623072, 1692768, 1896336, 1925196, 2024760, 2388720, 2529090, 2836008, 3423120, 3724320, 3822360, 4628760, 4750920, 7219608, 7359912, 7603488, 7749060

Offset: 1

Paolo P. Lava, Mar 25 2013

nonn

			47940 is in the sequence because sigma(47940)=145152, sigma(47941)=48384, and 145152=3*48384.
7749060 is in the sequence because sigma(7749060)=24192000, sigma(7749061)=8064000, and 24192000=3*8064000.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000203, A002961, A058073, A067081, A077087, A163193, A272027 (3*sigma(n)).

[n: n in [1..10^7] | SumOfDivisors(n) eq 3*SumOfDivisors(n+1)]; // Bruno Berselli, Mar 25 2013

A217791:=proc(q) local n;
for n from 1 to q do if sigma(n)=3*sigma(n+1) then print(n); fi; od; end:
A217791(10^10);

Position[Partition[DivisorSigma[1,Range[78*10^5]],2,1],?(#[[1]] == 3#[[2]]&), {1},Heads->False]//Flatten (* _Harvey P. Dale, Oct 17 2016 *)

More terms from Bruno Berselli, Mar 25 2013

A227304 Numbers k such that sigma(k+1) divides sigma(k-1).

Original entry on oeis.org

34, 55, 285, 367, 835, 849, 919, 1241, 1505, 2911, 2914, 3305, 4149, 4188, 6111, 6903, 7170, 7913, 8506, 9360, 10251, 10541, 12566, 15086, 17273, 17815, 19005, 19689, 21411, 21462, 24882, 25020, 25501, 26610, 28125, 30361, 30593, 30789, 31485, 37741, 38211, 38983, 39787

Offset: 1

Alex Ratushnyak, Jul 05 2013

nonn

The sequence consists mainly of terms of A055574 = { n | sigma(n+1) = sigma(n-1) }. - M. F. Hasler, Aug 06 2015

			Sigma(8507) = 8736 divides sigma(8505) = 17472 = 8736*2, so 8506 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..6000 (term 1..280 from Harvey P. Dale)

Cf. A000203, A055574, A067130, A058073.

Flatten[Position[Partition[DivisorSigma[1,Range[40000]],3,1], ?(Divisible[ #[[1]],#[[3]]]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Jul 15 2014 *)

A227240 Numbers k such that sigma(k) divides sigma(2k) and sigma(2k + 1).

Original entry on oeis.org

1, 3, 5, 7, 11, 23, 29, 41, 53, 77, 83, 89, 103, 113, 131, 143, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 533, 593, 641, 653, 659, 667, 683, 719, 743, 761, 807, 809, 817, 911, 953, 1013, 1019, 1031, 1049, 1073, 1103, 1223, 1229, 1289, 1409

Offset: 1

Alex Ratushnyak, Jul 03 2013

nonn

Numbers such that 2*k and/or 2*k + 1 is also in the sequence: 1, 3, 5, 11, 41, 89, 179, 359, 509, 719, 743, ... (Cf. A007700).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000203, A007700, A058072, A058073.

Select[Range[1000], IntegerQ[DivisorSigma[1, 2#]/DivisorSigma[1, #]] && IntegerQ[DivisorSigma[1, 2# + 1]/DivisorSigma[1, #]] &] (* Alonso del Arte, Jul 15 2013 *)
Select[Range[1500],And@@Divisible[{DivisorSigma[1,2#],DivisorSigma[1,2#+1]}, DivisorSigma[1,#]]&] (* Harvey P. Dale, Feb 25 2016 *)

isok(n) = my(sn=sigma(n)); !(sigma(2*n) % sn) && !(sigma(2*n+1) % sn); \\ Michel Marcus, Oct 02 2017

A340582 Numbers k such that sigma(k) = 4 * sigma(k+1).

Original entry on oeis.org

25908120, 136616760, 171430560, 196876680, 354049920, 405601560, 514374840, 825473880, 1204476000, 2223650520, 2510539920, 3003191100, 4017339480, 4574146500, 6129062940, 6797728800, 7311296520, 9779952000, 12472435080, 13103164800, 19989450000, 22180840920, 22710872520

Offset: 1

Seiichi Manyama, Jan 21 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..30 (using A058073 data)

Cf. A002961, A080372, A163193, A217791, A340713, A340720.

a(9)-a(23) from Seiichi Manyama using A058073 data, Jan 21 2021

A260988 Numbers n such that sigma(n) = m*sigma(n+2) with some m > 1.

Original entry on oeis.org

8505, 25500, 30360, 37740, 40740, 123795, 141480, 185460, 442365, 485400, 834435, 953568, 1055460, 1097820, 1108320, 1397220, 1953960, 2088480, 2208840, 2571744, 3050784, 3342816, 3544695, 3810456, 4314156, 4725660, 4867236, 5638776, 6318180, 6596340, 8428320, 8630832, 9347280, 9576336, 9908460, 10271580, 10992360, 11789925

Offset: 1

M. F. Hasler, Aug 06 2015

nonn

Sequence A227304 is the union of A055574 and { a(n)+1 }.

Many but not all terms are multiples of 5.

Cf. A000203, A055574, A067130, A058073.

for(n=1,2e9,sigma(n)%sigma(n+2)||sigma(n)==sigma(n+2)||print1(n","))

cp's OEIS Frontend

A163193 Numbers k such that sigma(k) = 2*sigma(k+1).

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A132585 Numbers k such that sigma(k)-k-1 divides sigma(k+1)-k-2, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

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A132586 Numbers k such that sigma(k+1)-k-2 divides sigma(k)-k-1, where sigma(k) is sum of positive divisors of k and the ratio is greater than zero.

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A217791 Numbers k such that sigma(k) = 3*sigma(k+1).

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A227304 Numbers k such that sigma(k+1) divides sigma(k-1).

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A227240 Numbers k such that sigma(k) divides sigma(2*k) and sigma(2*k + 1).

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A340582 Numbers k such that sigma(k) = 4 * sigma(k+1).

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A260988 Numbers n such that sigma(n) = m*sigma(n+2) with some m > 1.

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A227240 Numbers k such that sigma(k) divides sigma(2k) and sigma(2k + 1).