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.

A325637 Numbers k for which gcd(2k, sigma(k)) = 2k.

Original entry on oeis.org

6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Comments

Multiply-perfect numbers (A007691) k with an even abundancy index sigma(k)/k. - Amiram Eldar, Jun 26 2024

Links

Crossrefs

Cf. A000203, A033879, A224832, A325636, A325638, A325654.
Subsequences: A000396, A336702 (after its initial 1).
Subsequence of A007691.

Programs

  • PARI
    isA325637(n) = ((n+n)==gcd(n+n,sigma(n)));

Formula

a(n) = A224832(n)/2. - Amiram Eldar, Jun 26 2024

A224907 Numbers n such that the sum of reciprocals of even divisors of n > 1.

Original entry on oeis.org

24, 36, 40, 48, 60, 72, 80, 84, 96, 108, 112, 120, 132, 140, 144, 156, 160, 168, 176, 180, 192, 200, 204, 208, 216, 224, 228, 240, 252, 264, 276, 280, 288, 300, 312, 320, 324, 336, 348, 352, 360, 372, 384, 392, 396, 400, 408, 416, 420, 432, 440, 444, 448, 456
Offset: 1

Views

Author

Michel Lagneau, Jul 25 2013

Keywords

Comments

Numbers n such that the sum of reciprocals of even divisors of n equals m/n for some integer m where the fraction m/n > 1. The corresponding numerators m are given by the sequence A204822(n) = {28, 39, 42, 60, 72, 91, 90, 96,...} (Sum of divisors (A000203) of abundant numbers (A005101)).

Examples

			40 is in the sequence because the even divisors of 40 are 2, 4, 8, 10, 20, 40 and 1/2 + 1/4 + 1/8 + 1/10 + 1/20 + 1/40 = 42/40 = A204823(3)/a(3), and 42/40 > 1.

Links

Crossrefs

Cf. A000203, A005101, A204822, A224832, A225241.

Programs

  • Maple
    ***program 1 where sum of reciprocals even divisors > 1***
with(numtheory):for n from 2 by 2 to 500 do:x:=divisors(n):n1:=nops(x): s:=0:for i from 1 to n1 do: if irem(x[i],2)=0 then s:=s+1/x[i]:else fi:od: if s>1 then printf(`%d, `,n):else fi:od:
***program 2 where sum of reciprocals even divisors = m/n***
with(numtheory):for n from 2 to 500 do:x:=divisors(n):n1:=nops(x): s:=0:for i from 1 to n1 do: if irem(x[i],2)=0 then s:=s+1/x[i]:else fi:od: for m from n+1 to 2*n do: if s=m/n then printf(`%d, `,n):else fi:od:od:
  • Mathematica
    Select[Range[500],Total[1/Select[Divisors[#],EvenQ]]>1&] (* Harvey P. Dale, Aug 15 2015 *)

Formula

a(n) = 2*A005101(n).

A225241 Numbers n such that the sum of the reciprocals of the even divisors of n is greater than zero and less than one.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 16, 18, 20, 22, 26, 28, 30, 32, 34, 38, 42, 44, 46, 50, 52, 54, 58, 62, 64, 66, 68, 70, 74, 76, 78, 82, 86, 88, 90, 92, 94, 98, 100, 102, 104, 106, 110, 114, 116, 118, 122, 124, 126, 128, 130, 134, 136, 138, 142, 146, 148, 150, 152, 154
Offset: 1

Views

Author

Michel Lagneau, Jul 25 2013

Keywords

Comments

Numbers n such that the sum of reciprocals of even divisors of n equals m/n for some integer m where the fraction m/n < 1.
The corresponding numerators m are given by the sequence A204823(n) = {1, 3, 4, 7, 6, 8, 15, 13, 18, 12, 14, 24,...} (Sum of divisors (A000203) of deficient numbers (A005100)).

Examples

			8 is in the sequence because the even divisors of 8 are 2, 4, 8 and 1/2 + 1/4 + 1/8 = 7/8 = A204823(4)/a(4).

Crossrefs

Cf. A000203, A005100, A204823, A224832, A224907.

Programs

  • Maple
    ***program 1 where sum of reciprocals even divisors < 1***
with(numtheory):for n from 2 by 2 to 500 do:x:=divisors(n):n1:=nops(x): s:=0:for i from 1 to n1 do: if irem(x[i], 2)=0 then s:=s+1/x[i]:else fi:od: if s<1 then printf(`%d, `, n):else fi:od:
***program 2 where sum of reciprocals even divisors = m/n***
with(numtheory):for n from 2 to 500 do:x:=divisors(n):n1:=nops(x): s:=0:for i from 1 to n1 do: if irem(x[i],2)=0 then s:=s+1/x[i]:else fi:od: for m from 1 to n-1 do: if s=m/n then printf(`%d, `,n):else fi:od:od:
  • Mathematica
    Select[Range[200],0Harvey P. Dale, Jan 10 2024 *)

Formula

a(n) = 2*A005100(n) where A005100 are deficient numbers: numbers n such that sigma(n) < 2n.

Extensions

Definition corrected by Harvey P. Dale, Jan 10 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.