cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 23 2007

Keywords

Comments

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

Examples

			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)

Crossrefs

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

Programs

  • Maple
    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);

Extensions

a(6) from Donovan Johnson, Aug 31 2008
a(7) by Jud McCranie, Jul 26 2025

A132630 Numbers n such that sigma(n)-n divides sigma(n+1)-n-1, where sigma(n) is sum of positive divisors of n.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 27 2007

Keywords

Comments

With the exception of the first term, only odd numbers. All the prime numbers p are included because sigma(p)-p=1.

Examples

			n=85 -> sigma(n+1)-n-1=1+2+43=46 sigma(n)-n=1+5+17=23 -> 46/23=2
n=125 -> sigma(n+1)-n-1=1+2+3+6+7+9+14+18+21+42+63=186 sigma(n)-n=1+5+25=31 -> 186/31=6

Links

Crossrefs

Cf. A132585, A132586, A132631.

Programs

  • Magma
    [k:k in [2..200]| IsIntegral((DivisorSigma(1,k+1)-k-1)/ (DivisorSigma(1,k)-k))]; // Marius A. Burtea, Nov 06 2019
  • Maple
    with(numtheory); P:=proc(n) local a,i; for i from 1 by 1 to n do if sigma(i)-i>0 then a:=(sigma(i+1)-i-1)/(sigma(i)-i); if a>0 and trunc(a)=a then print(i); fi; fi; od; end: P(200)
  • Mathematica
    Select[Range[2,200],Divisible[DivisorSigma[1,#+1]-#-1,DivisorSigma[ 1,#]-#]&] (* Harvey P. Dale, Apr 25 2015 *)

Extensions

Comment amended by Harvey P. Dale, Apr 25 2015

A132631 Numbers k such that sigma(k+1)-k-1 divides sigma(k)-k, where sigma(k) is sum of positive divisors of n.

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 18, 20, 22, 24, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 94, 96, 100, 102, 106, 108, 112, 120, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 27 2007

Keywords

Comments

Only even numbers.

Examples

			k=94 -> sigma(k)-k=1+2+47=50 sigma(k+1)-k-1=1+5+19=25 -> 50/25=2
k=120 -> sigma(k)-k=1+2+3+4+5+6+8+10+12+15+20+24+30+40+60=240 sigma(k+1)-k-1=1+11=12 -> 240/12=20

Links

Crossrefs

Cf. A132585, A132586, A132630.

Programs

  • Maple
    with(numtheory): P:=proc(k) if frac((sigma(k)-k)/(sigma(k+1)-k-1))=0 then k; fi; end: seq(P(n),n=2..200);
  • Mathematica
    Select[Range[2,200],Divisible[DivisorSigma[1,#]-#,DivisorSigma[1,#+1]-#-1]&] (* Harvey P. Dale, May 20 2017 *)
Showing 1-3 of 3 results.

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