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 1-5 of 5 results.

A064591 Nonunitary perfect numbers: k is the sum of its nonunitary divisors; i.e., k = sigma(k) - usigma(k).

Original entry on oeis.org

24, 112, 1984, 32512, 134201344, 34359476224, 549754765312
Offset: 1

Views

Author

Dean Hickerson, Sep 25 2001

Keywords

Comments

There are no other terms up to 1.2*10^14.
If P (A000396) is an even perfect number, then 4*P is in the sequence. Are there any others?
If there are no terms of another form, the sequence goes on with 9223372032559808512 = 2^32 * A000668(8), 10633823966279326978618770463815368704 = 2^62 * A000668(9), 766247770432944429179173512337214552523989286192676864 = 2^90 * A000668(10), ... - Michel Lagneau, Jan 21 2015
Conjecture: let s0 be the sum of the inverses of the even divisors of a number n and s1 the sum of the inverses of the odd divisors of n; then n is in the sequence iff 2*s0-s1 = 1. - Michel Lagneau, Jan 21 2015
Ligh & Wall proved that 2^(p+1)*(2^p-1) is a term if p and 2^p-1 are primes, and that all the nonunitary perfect numbers below 10^6 are of this form. - Amiram Eldar, Sep 27 2018
If k is in the sequence and k = 2^m*p^a then k is of the form 4*P for an even perfect P. See the link to MathOverflow. - Joshua Zelinsky, Mar 07 2024

Examples

			The sum of the nonunitary divisors of 24 is 2 + 4 + 6 + 12 = 24.

Links

Crossrefs

Cf. A000203 (sigma), A034448 (usigma), A048146, A063870.
Cf. A064592, A064593, A064594, A064595, A064596, A064597, A064598, A064599.

Programs

  • Mathematica
    nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]==n, Print[ n ] ] ]
Do[s0=0;s1=0;Do[d=Divisors[n][[i]];If[Mod[d,2]==0,s0=s0+1/d,s1=s1+1/d],{i,1,Length[Divisors[n]]}];If[2*s0-s1==1,Print[n]],{n,2,10^9,2}] (* Michel Lagneau, Jan 21 2015 *)

A064594 Nonunitary multiply perfect numbers: the sum of the nonunitary divisors of n is a multiple of n; i.e., n divides sigma(n) - usigma(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109
Offset: 1

Views

Author

Dean Hickerson, Sep 25 2001

Keywords

Comments

Trivially includes all squarefree numbers (A005117). See A064595 for the others.

Links

Crossrefs

Cf. A048146, A064591, A064592, A064593, A064595, A064596.

Programs

  • Mathematica
    nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ Mod[ nusigma[ n ], n ]==0, Print[ n ] ] ]
  • PARI
    usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; for (m=1, 10^9, if ((sigma(m) - usigma(m)) % m == 0, write("b064594.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 19 2009

A064595 Nontrivial nonunitary multiply perfect numbers: the sum of the nonunitary divisors of n is a positive multiple of n; i.e., (sigma(n) - usigma(n))/n is a positive integer.

Original entry on oeis.org

24, 112, 1984, 2520, 31320, 32512, 1163160, 2208384, 3053232, 6535296, 13472928, 123165600, 134201344, 186076800, 365959296, 3456573120, 4401782352, 5517818880, 18924364800, 34359476224, 325377803520
Offset: 1

Views

Author

Dean Hickerson, Sep 25 2001

Keywords

Crossrefs

Cf. A048146, A064591, A064592, A064593, A064594, A064596.

Programs

  • Mathematica
    nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, v=nusigma[ n ]; If[ v>0&&Mod[ v, n ]==0, Print[ n ] ] ]

A064592 Nonunitary doubly perfect numbers: the sum of the nonunitary divisors of n is 2n; i.e., sigma(n) - usigma(n) = 2n.

Original entry on oeis.org

2520, 31320, 1163160, 2208384, 3053232, 6535296, 13472928, 123165600, 365959296, 4401782352, 5517818880, 612014161920
Offset: 1

Views

Author

Dean Hickerson, Sep 25 2001

Keywords

Comments

There are no others up to 7.9*10^12, but here's a larger one: 2846972800010800140288.
Ligh & Wall found the first 11 terms of this sequence. - Amiram Eldar, Sep 27 2018

Links

Crossrefs

Cf. A048146, A064591, A064593, A064594, A064595, A064596.

Programs

  • Mathematica
    nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]==2n, Print[ n ] ] ]
  • PARI
    isok(n) = sumdiv(n, d, if (gcd(d, n/d)!=1, d)) == 2*n; \\ Michel Marcus, Jul 31 2017

A064593 Nonunitary triply perfect numbers: the sum of the nonunitary divisors of n is 3n; i.e., sigma(n) - usigma(n) = 3n.

Original entry on oeis.org

186076800, 3456573120, 18924364800, 325377803520, 2666567816640, 8538872307840, 11169839116800
Offset: 1

Views

Author

Dean Hickerson, Sep 25 2001

Keywords

Comments

There are no others up to 1.6*10^13, but there are at least 20 larger ones, between 87916260096000 and 19424258892617980729341453189120.
Ligh & Wall found a(1), a(4) and a(5). - Amiram Eldar, Sep 27 2018

Links

Crossrefs

Cf. A048146, A064591, A064592, A064594, A064595, A064596.
Showing 1-5 of 5 results.

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