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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.

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A005820 3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

Original entry on oeis.org

120, 672, 523776, 459818240, 1476304896, 51001180160
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

These six terms are believed to comprise all 3-perfect numbers. - cf. the MathWorld link. - Daniel Forgues, May 11 2010
If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, since sigma(2m) = sigma(2)*sigma(m) = 3*2m. - Jens Kruse Andersen, Jul 30 2014
According to the previous comment from Jens Kruse Andersen, proving that this sequence is complete would imply that there are no odd perfect numbers. - Farideh Firoozbakht, Sep 09 2014
If 2 were prepended to this sequence, then it would be the sequence of integers k such that numerator(sigma(k)/k) = A017665(k) = 3. - Michel Marcus, Nov 22 2015
From Antti Karttunen, Mar 20 2021, Sep 18 2021, (Start):
Obviously, any odd triperfect numbers k, if they exist, have to be squares for the condition sigma(k) = 3*k to hold, as sigma(k) is odd only for k square or twice a square. The square root would then need to be a term of A097023, because in that case sigma(2*k) = 9*k. (See illustration in A347391).
Conversely to Jens Kruse Andersen's comment above, any 3-perfect number of the form 4k+2 would be twice an odd perfect number. See comment in A347870.
(End)

Examples

			120 = 2^3*3*5;  sigma(120) = (2^4-1)/1*(3^2-1)/2*(5^2-1)/4 = (15)*(4)*(6) = (3*5)*(2^2)*(2*3) = 2^3*3^2*5 = (3) * (2^3*3*5) = 3 * 120. - _Daniel Forgues_, May 09 2010

References

  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 120, p. 42, Ellipses, Paris 2008.
  • R. K. Guy, Unsolved Problems in Number Theory, B2.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chap.15, pp 82-5, Belin/Pour la Science, Paris 2000.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 142.
  • David Wells, "The Penguin Book of Curious and Interesting Numbers," Penguin Books, London, 1986, pages 135, 159 and 185.

Links

Crossrefs

Cf. A000203, A000396, A007539, A017665, A019278, A027687, A046060, A046061, A068403, A075701, A097023, A171266, A259302, A259303, A306373, A326051, A326181, A329189, A335141, A335254, A347383, A347391.
Subsequence of the following sequences: A007691, A069085, A153501, A216780, A292365, A336458, A336461, A336745, and if there are no odd terms, then also of A334410.
Positions of 120's in A094759, 119's in A326200.

Programs

Formula

a(n) = 2*A326051(n). [provided no odd triperfect numbers exist] - Antti Karttunen, Jun 13 2019

Extensions

Wells gives the 6th term as 31001180160, but this is an error.
Edited by Farideh Firoozbakht and N. J. A. Sloane, Sep 09 2014 to remove some incorrect statements.

A094759 Least k <= n such that n*sigma(k) = k*sigma(n), where sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74
Offset: 1

Views

Author

Amarnath Murthy, May 30 2004

Keywords

Comments

Conjecture: There are infinitely many terms such that a(n)A050973 has those n, A050972 has the a(n).
See A095301 for a version of A050973 that do not duplicate every n that has several smaller k of the same abundancy. - Jeppe Stig Nielsen, Jul 09 2015
That conjecture is an easy fact: Since, e.g., (6,28) is a friendly pair, then so is (6k,28k) for any multiplier k with gcd(42,k)=1. So any n=28k, where gcd(42,k)=1, satisfies a(n)A095301 does not have asymptotic density zero. - Jeppe Stig Nielsen, Jul 09 2015
This sequence is related to Theorem 1 on p. 173 of the Erdős link in the following way. For a given x, let us consider the set of integers such that a(n) <= x, which is equivalent to removing duplicates from the current sequence. This set would begin with: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, ... So this set has the same number of elements as the number of distinct terms numbers of the form sigma(n)/n with 1 <= n <=x. Then by Erdős, it is c1*x + o(x), with 6/Pi^2 < c1 < 1. With x = 10^7, we find c1 ~= 0.98208... - Michel Marcus, Jul 21 2015
a(n) is the least k which has the same abundancy index as n, that is, minimal k for which sigma(k)/k = sigma(n)/n. - Antti Karttunen, Jul 10 2019

Links

Crossrefs

Cf. A000203, A050972, A050973, A094757, A094758, A326200.
Cf. A095301 for n such that a(n) < n.
Cf. A000396 (positions of 6's), A005820 (positions of 120's).

Programs

  • Maple
    N:= 100: # to get a(1) to a(N)
for n from 1 to N do
   v:= numtheory:-sigma(n)/n;
   if not assigned(R[v]) then R[v]:= n fi;
   A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, Jul 21 2015
  • Mathematica
    Table[Module[{k=1,sn=DivisorSigma[1,n]},While[n DivisorSigma[1,k]!=k*sn,k++];k],{n,80}] (* Harvey P. Dale, Aug 03 2025 *)
  • PARI
    for(n=1,74,s=sigma(n);k=1;while(n*sigma(k)!=k*s,k++);print1(k,","));

Extensions

Edited and extended by Don Reble and Klaus Brockhaus, May 31 2004

A326202 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 2 and gcd(n,sigma(n)) = 1, with f(n) = n for all other numbers.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 3, 3, 3, 5, 3, 6, 3, 7, 8, 3, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 14, 3, 15, 3, 3, 16, 17, 3, 3, 3, 18, 3, 19, 3, 20, 3, 21, 22, 23, 3, 24, 3, 3, 25, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 3, 3, 32, 3, 33, 34, 35, 3, 36, 3, 37, 3, 38, 3, 39, 3, 40, 3, 41, 3, 42, 3, 43, 44, 45, 3, 46, 47, 48, 3, 49, 50, 51, 3, 3, 52, 3, 3, 53, 3, 54, 55
Offset: 1

Views

Author

Antti Karttunen, Jul 11 2019

Keywords

Comments

For all i, j:
A305801(i) = A305801(j) => A324401(i) = A324401(j) => a(i) = a(j),
a(i) = a(j) => A009194(i) = A009194(j) => A325964(i) = A325964(j).

Links

Crossrefs

Cf. A000203, A009194, A014567, A305801, A324401, A325964.
Cf. also A326200, A326201, A326203.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux326202(n) = if((n>2) && (1==gcd(n,sigma(n))),0,n);
v326202 = rgs_transform(vector(up_to, n, Aux326202(n)));
A326202(n) = v326202[n];
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