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.

A066289 Numbers k such that k divides DivisorSigma(2*j-1, k) for all j; i.e., all odd-power-sums of divisors of k are divisible by k.

Original entry on oeis.org

1, 6, 120, 672, 30240, 32760, 31998395520, 796928461056000, 212517062615531520, 680489641226538823680000, 13297004660164711617331200000, 1534736870451951230417633280000, 6070066569710805693016339910206758877366156437562171488352958895095808000000000
Offset: 1

Views

Author

Labos Elemer, Dec 12 2001

Keywords

Comments

Tested for each k and j < 200. Otherwise the proof for all j seems laborious, since the number of divisors of terms of sequence rapidly increases: {1, 4, 16, 24, 96, 96, 2304, ...}.
Tested for each k and j <= 1000. - Thomas Baruchel, Oct 10 2003
The given terms have been tested for all j. - Don Reble, Nov 03 2003
This is a proper subset of the multiply perfect numbers A007691. E.g., 8128 from A007691 is not here because its remainder at Sigma[odd,8128]/8128 division is 0 or 896 depending on odd exponent.

Crossrefs

Cf. A066135, A066284, A007691, A066290.

Formula

DivisorSigma(2*j-1, k)/k is an integer for all j = 1, 2, 3, ..., 200, ...

Extensions

The following numbers belong to the sequence, but there may be missing terms in between: 796928461056000 (also belongs to A046060); 212517062615531520 (also belongs to A046060); 680489641226538823680000 (also belongs to A046061); 13297004660164711617331200000 (also belongs to A046061). - Thomas Baruchel, Oct 10 2003
Extended to 13 confirmed terms by Don Reble, Nov 04 2003. There is a question whether there are other members below a(13). However, there are none in Achim's list of multiperfect numbers (see A007691); Richard C. Schroeppel has suggested that that list is complete to 10^70 - if so, a(1..12) are correct; as for a(13), Rich says there's only "an epsilon chance that some undiscovered MPFN lies in the gap." So it is very likely to be correct. - Don Reble

A066292 Numbers n such that n divides sigma_(2^k)(n), the sum of the 2^k powers of the divisors of n, for all k>1.

Original entry on oeis.org

1, 84, 156, 364, 1092, 435708, 986076, 1118480, 1441188, 1674036, 2446668, 2597868, 3108924, 3875508, 4150692, 5537196, 6066396, 6686316, 13729212, 14639436, 18735444, 23307732, 27092052, 31806684, 58266468, 69728724
Offset: 1

Views

Author

Labos Elemer, Dec 12 2001

Keywords

Comments

Let d be the vector of divisors of n. The sequence d^(2^k) mod n has some period p. Thus if n divides sigma_(2^k)(n) for one period, then n divides sigma_(2^k)(n) for all k. For these n, the first period ends for k < 158. Hence it is easy to verify divisibility for all k. - T. D. Noe, Apr 11 2006

Examples

			n=84 is here because 84 divides each one of sigma_4(n)=53771172, sigma_8(n)=2488859101224132, sigma_16(n)=6144339637187846520573009496452, etc.

Crossrefs

Cf. A066135, A066284, A066289-A066292.
Cf. A118076.

Programs

  • Mathematica
    t={}; Do[If[Mod[DivisorSigma[4,n],n]==0, AppendTo[t,n]], {n,10^8}]; Do[t=Select[t,Mod[DivisorSigma[2^k,# ],# ]==0&],{k,3,20}]; t (* T. D. Noe, Apr 11 2006 *)

Extensions

Edited by T. D. Noe, Apr 11 2006

A066290 Numbers m such that DivisorSigma(4*k-2, m) mod m = 0 holds presumably for all k; that is, (4k-2)-power-sums of divisors of m are divisible by m for all k.

Original entry on oeis.org

1, 10, 60, 65, 130, 150, 260, 780, 1105, 2210, 4420, 8840, 13260, 19720, 20737, 32045, 41474, 55250, 64090, 82948, 103685, 128180, 207370, 207553, 221000, 248844, 256360, 295800, 331500, 352529, 384540, 414740, 415106, 450840, 512720, 705058, 829480, 830212
Offset: 1

Views

Author

Labos Elemer, Dec 12 2001

Keywords

Examples

			Tested for each m and k < 200. Proof for several values of k seems not so tedious because the number of divisors of the terms of the sequence is not so large: {1, 4, 12, 4, 8, 12, 12, 24, 8, 16, 24, 32, 48, 32, 4, 16, 8, 32, 32, 12, 8, 48, 16, 8, 64, 24, 64, 96, 96, 8, 96, 24, 16, 96, 80, 16, 32, 24}.

Crossrefs

Cf. A066135, A066284, A066289.

Programs

  • Mathematica
    lastSeq = {}; max = 100; While[seq = Reap[For[n = 1, n < 10^6, n++, If[AllTrue[Range[max], Mod[DivisorSigma[4 # - 2, n], n] == 0&], Print[n]; Sow[n]]]][[2, 1]]; seq != lastSeq, lastSeq = seq; max = max + 100; Print["max = ", max]]; seq (* Jean-François Alcover, Oct 02 2016 *)

Formula

DivisorSigma(4k-2, m)/m is an integer for k = 1, 2, 3, .., 200, ...

Extensions

More terms from Jean-François Alcover, Oct 02 2016

A066291 Numbers m such that DivisorSigma(8*k-4, m) mod m = 0 holds presumably for all k; that is, (8*k-4)-power-sums of divisors of m are divisible by m for all k.

Original entry on oeis.org

1, 34, 492, 5617, 11234, 22468, 67404, 190978, 709937, 763912, 1419874, 2839748, 5073996, 5446841, 7914353, 8519244, 10893682, 11548552, 15828706, 17126233, 21787364, 31657412, 34252466, 43574728, 57928121, 63314824, 65362092, 68504932, 73084632, 94972236
Offset: 1

Views

Author

Labos Elemer, Dec 12 2001

Keywords

Examples

			Tested for each m with k < 200.
Tested for each m with k < 500. - _Sean A. Irvine_, Oct 07 2023

Crossrefs

Cf. A066135, A066284, A066289-A066292.

Programs

  • Mathematica
    Table[Union[Table[ IntegerQ[DivisorSigma[8*k-4, Part[t, m]]/Part[t, m]], {k, 1, 200}]], {m, 1, Length[t]}]; where t denotes the table of sequence.

Formula

DivisorSigma(8*k-4, m)/m is an integer for k = 1, 2, 3, ..., 200, ...

Extensions

More terms from Sean A. Irvine, Oct 07 2023
Showing 1-4 of 4 results.

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