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 1-4 of 4 results.

A204822 Sum of divisors (A000203) of abundant numbers (A005101).

Original entry on oeis.org

28, 39, 42, 60, 72, 91, 90, 96, 124, 120, 120, 168, 144, 144, 195, 168, 186, 224, 180, 234, 252, 217, 216, 210, 280, 248, 240, 360, 312, 336, 288, 336, 403, 372, 392, 378, 363, 480, 360, 372, 546, 384, 508, 399, 468, 465, 504, 434, 576, 600, 504, 456, 504, 560
Offset: 1

Views

Author

Jaroslav Krizek, Jan 22 2012

Keywords

Links

Crossrefs

Cf. A204823 (sum of divisors of deficient numbers), A205098 (sum of proper divisors of abundant numbers).

Programs

  • Mathematica
    sdan[n_]:=Module[{ds=DivisorSigma[1,n]},If[ds>2n,ds,0]]; Select[ Array[ sdan, 300],#>0&] (* Harvey P. Dale, Aug 15 2015 *)
  • PARI
    for(n=6,200,s=sigma(n);if(s>2*n,print1(s", "))) \\ Charles R Greathouse IV, Feb 19 2013

Formula

a(n) = A000203(A005101(n)) = A005101(n) + A205098(n).
a(n) << n log log n with lim sup a(n)/(n log log n) approximately 7.192. - Charles R Greathouse IV, Feb 19 2013

A205099 Sum of proper divisors (A001065) of deficient numbers (A005100).

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 7, 4, 8, 1, 1, 10, 9, 15, 1, 1, 11, 14, 1, 6, 16, 13, 1, 1, 31, 15, 20, 13, 1, 22, 17, 1, 1, 40, 33, 26, 1, 8, 43, 21, 46, 1, 17, 23, 32, 1, 1, 34, 41, 63, 19, 1, 58, 27, 1, 1, 40, 49, 64, 19, 1, 40, 44, 1, 23, 46, 33, 1, 21, 76, 35, 50
Offset: 1

Views

Author

Jaroslav Krizek, Jan 22 2012

Keywords

Links

Crossrefs

Cf. A205098 (sum of proper divisors of abundant numbers), A204823 (sum of divisors of deficient numbers).

Programs

Formula

a(n) = 1 for A005100(n) = prime.
a(n) = A001065(A005100(n)) = A204823(n) - A005100(n).

Extensions

Spurious term removed by Jaroslav Krizek, Nov 26 2014

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 1-4 of 4 results.

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