A058072 -id:A058072 - cp's OEIS frontend

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.

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

A340713 Numbers k such that sigma(k+1) = 4 * sigma(k).

Original entry on oeis.org

37033919, 141162839, 264995639, 596672999, 606523679, 630777839, 791656319, 920424119, 1060332839, 1379454719, 1954690919, 3799661039, 4024838999, 4633959959, 5393988599, 5935994063, 8831231639, 9866482079, 11237657759, 11273710139, 12266364599, 14440498379, 14952625379

Offset: 1

Seiichi Manyama, Jan 17 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..40 (using A058072 data)

Cf. A000203, A002961, A058072, A067081, A077087, A340715.

isok(k) = sigma(k+1) == 4*sigma(k); \\ Michel Marcus, Jan 18 2021

a(10)-a(23) from Seiichi Manyama using A058072 data, Jan 17 2021

A340715 Least positive number k such that sigma(k+1) = n * sigma(k).

Original entry on oeis.org

14, 5, 1, 37033919, 14182439039

Offset: 1

Seiichi Manyama, Jan 17 2021

			  n | sigma(a(n)) | sigma(a(n)+1)
----+-------------+--------------
  1 |          24 |            24
  2 |           6 |            12
  3 |           1 |             3
  4 |    39940992 |     159763968
  5 | 14182439040 |   70912195200

Cf. A000203, A002961, A058072, A067081, A077087, A080371, A340713, A340720.

k = 1;n = 1;Print[While[DivisorSigma[1, k + 1] != n*DivisorSigma[1, k], k;k k+]; k] (* Robert P. P. McKone, Jan 17 2021 *)

{a(n) = my(k=1); while(sigma(k+1)!=n*sigma(k), k++); k}

A077089 Quotients when sigma(k+1)/sigma(k) is an integer.

Original entry on oeis.org

3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 3, 1, 2, 1, 2, 2, 3, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 3, 2, 2, 1, 2, 2, 1, 2, 3, 3, 1, 3, 1, 2, 2, 2, 3, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 3, 2, 1, 2, 1, 2, 2, 3, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Oct 31 2002

nonn

			a(1) = sigma(2)/sigma(1) = 3/1 = 3.
a(2) = sigma(6)/sigma(5) = 12/6 = 2.
a(3) = sigma(15)/sigma(14) = 24/24 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..500 from Seiichi Manyama)

Cf. A000203, A002961, A058072, A067081, A077086, A077087.

Do[s=Mod[a=DivisorSigma[1, n+1], b=DivisorSigma[1, n]]; If[Equal[s, 0], Print[a/b]], {n, 1, 10000000}]
Select[#[[2]]/#[[1]]&/@Partition[DivisorSigma[1,Range[10^6]],2,1], IntegerQ] (* Harvey P. Dale, Dec 26 2015 *)

a(n) = sigma(A058072(n)+1)/sigma(A058072(n)). - Seiichi Manyama, Jan 16 2021

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

cp's OEIS Frontend

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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A340713 Numbers k such that sigma(k+1) = 4 * sigma(k).

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A340715 Least positive number k such that sigma(k+1) = n * sigma(k).

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A077089 Quotients when sigma(k+1)/sigma(k) is an integer.

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

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