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 1-8 of 8 results.

A007691 Multiply-perfect numbers: n divides sigma(n).

Original entry on oeis.org

1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 1476304896, 8589869056, 14182439040, 31998395520, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608
Offset: 1

Views

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

Comments

sigma(n)/n is in A054030.
Also numbers such that the sum of the reciprocals of the divisors is an integer. - Harvey P. Dale, Jul 24 2001
Luca's solution of problem 11090, which proves that for k>1 there are an infinite number of n such that n divides sigma_k(n), does not apply to this sequence. However, it is conjectured that this sequence is also infinite. - T. D. Noe, Nov 04 2007
Numbers k such that sigma(k) is divisible by all divisors of k, subsequence of A166070. - Jaroslav Krizek, Oct 06 2009
A017666(a(n)) = 1. - Reinhard Zumkeller, Apr 06 2012
Bach, Miller, & Shallit show that this sequence can be recognized in polynomial time with arbitrarily small error by a probabilistic Turing machine; that is, this sequence is in BPP. - Charles R Greathouse IV, Jun 21 2013
Conjecture: If n is such that 2^n-1 is in A066175 then a(n) is a triangular number. - Ivan N. Ianakiev, Aug 26 2013
Conjecture: Every multiply-perfect number is practical (A005153). I've verified this conjecture for the first 5261 terms with abundancy > 2 using Achim Flammenkamp's data. The even perfect numbers are easily shown to be practical, but every practical number > 1 is even, so a weak form says every even multiply-perfect number is practical. - Jaycob Coleman, Oct 15 2013
Numbers such that A054024(n) = 0. - Michel Marcus, Nov 16 2013
Numbers n such that k(n) = A229110(n) = antisigma(n) mod n = A024816(n) mod n = A000217(n) mod n = (n(n+1)/2) mod n = A142150(n). k(n) = n/2 for even n; k(n) = 0 for odd n (for number 1 and eventually odd multiply-perfect numbers n > 1). - Jaroslav Krizek, May 28 2014
The only terms m > 1 of this sequence that are not in A145551 are m for which sigma(m)/m is not a divisor of m. Conjecture: after 1, A323653 lists all such m (and no other numbers). - Antti Karttunen, Mar 19 2021

Examples

			120 is OK because divisors of 120 are {1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120}, the sum of which is 360=120*3.

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 22.
  • J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 176.
  • 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", Chapter 15, pp. 82-88, Belin-Pour La Science, Paris 2000.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 141-148.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 135-136.

Links

Crossrefs

Complement is A054027. Cf. A000203, A054030.
Cf. A000396, A005820, A027687, A046060, A046061, for subsequences of terms with quotient sigma(n)/n = 2..6.
Other subsequences: A046985, A046986, A046987, A047728, A065997, A066289, (A076231, A076233, A076234), A088844, A088845, A088846, A091443, A114887, A166069, A245782, A260508, A306667, (A325021 U A325022), (A325023 U A325024), (A325025 U A325026), A325637, A323653, A330532, (A330533 U A331724), A336702, A341045.
Subsequence of the following sequences: A011775, A071707, A083865, A089748 (after the initial 1), A102783, A166070, A175200, A225110, A226476, A237719, A245774, A246454, A259307, A263928, A282775, A323652, A336745, A340864. Also conjectured to be a subsequence of A005153, of A307740, and after 1 also of A295078.
Various number-theoretical functions applied to these numbers: A088843 [tau], A098203 [phi], A098204 [gcd(a(n),phi(a(n)))], A134665 [2-adic valuation], A307741 [sigma], A308423 [product of divisors], A320024 [the odd part], A134740 [omega], A342658 [bigomega], A342659 [smallest prime not dividing], A342660 [largest prime divisor].
Positions of ones in A017666, A019294, A094701, A227470, of zeros in A054024, A082901, A173438, A272008, A318996, A326194, A341524. Fixed points of A009194.
Cf. A069926, A330746 (left inverses, when applied to a(n) give n).
Cf. A007358, A189000, A327158, A332318/A332319 (for analogs) and A046762, A046763, A046764, A055715, A056006, A081756, A214842, A227302, A227306, A245775, A300906, A325639 (other variants).
Cf. (other related sequences) A007539, A066135, A066961, A093034, A094467, A134639, A145551, A019278, A194771 [= 2*a(n)], A219545, A229110, A262432, A335830, A336849, A341608.

Programs

  • Haskell
    a007691 n = a007691_list !! (n-1)
a007691_list = filter ((== 1) . a017666) [1..]
-- Reinhard Zumkeller, Apr 06 2012
  • Mathematica
    Do[If[Mod[DivisorSigma[1, n], n] == 0, Print[n]], {n, 2, 2*10^11}] (* or *)
Transpose[Select[Table[{n, DivisorSigma[-1, n]}, {n, 100000}], IntegerQ[ #[[2]] ]& ] ][[1]]
(* Third program: *)
Select[Range[10^6], IntegerQ@ DivisorSigma[-1, #] &] (* Michael De Vlieger, Mar 19 2021 *)
  • PARI
    for(n=1,1e6,if(sigma(n)%n==0, print1(n", ")))
  • Python
    from sympy import divisor_sigma as sigma
def ok(n): return sigma(n, 1)%n == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

Extensions

More terms from Jud McCranie and then from David W. Wilson.
Incorrect comment removed and the crossrefs-section reorganized by Antti Karttunen, Mar 20 2021

A166070 Sorted sequence of primes and multiply perfect numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 23, 28, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263
Offset: 1

Views

Author

Jaroslav Krizek, Oct 06 2009, Oct 16 2009

Keywords

Comments

Numbers k such that sigma(k) is divisible by all proper divisors of k.

Examples

			a(4) = 6: all proper divisors of 6 (1, 2, 3) divide sigma(6) = 12.

Links

Crossrefs

Cf. A053813.

Programs

  • Mathematica
    Select[Range[300],And@@Divisible[DivisorSigma[1,#],Most[Divisors[#]]]&] (* Harvey P. Dale, Jan 18 2015 *)

Formula

A000040 Union A007691.
{1} Union A053813 Union A166069.
{1} Union A000040 Union A000396 Union A166069.

A371920 Abundant numbers whose abundance is also an abundant number.

Original entry on oeis.org

24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 112, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 176, 180, 186, 198, 204, 208, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282, 294, 304, 306, 308, 312, 318, 330, 336, 342, 348, 354, 360
Offset: 1

Views

Author

Amiram Eldar, Apr 12 2024

Keywords

Comments

First differs from A125639 at n = 12.
Numbers k such that A033880(k) > 0 and A033880(A033880(k)) > 0.
This sequence is infinite: if m is divisible by 6 and coprime to 5, then 5*m is a term.
All the multiply-perfect numbers (A007691) that are not 1 or perfect (A000396), i.e., the terms of A166069, are terms of this sequence.

Examples

			24 is a term since A033880(24) = 12 > 0 and A033880(12) = 4 > 0.

Links

Crossrefs

Cf. A033880 (abundance), A000396, A007691, A125639.
Subsequence of A005101.
Subsequences: A141545, A166069, A223610, A223611, A223613.

Programs

  • Mathematica
    ab[n_] := DivisorSigma[1, n] - 2*n; q[n_] := Module[{k = ab[n]}, k > 0 && ab[k] > 0]; Select[Range[360], q]
  • PARI
    ab(n) = sigma(n) - 2*n;
is(n) = {my(k = ab(n)); k > 0 && ab(k) > 0;}

A183020 Largest members of k-sociable cycles of order r, with k > 1 and r > 1.

Original entry on oeis.org

8132064, 14246719968, 97998179400, 582340505600
Offset: 1

Views

Author

William Rex Marshall, Jan 08 2011

Keywords

Comments

A k-sociable (or multisociable) cycle of order r consists of r distinct positive integers such that the sum of the aliquot divisors (or proper divisors) of each is equal to k times the next term in the cycle, where k (the multiplicity) is a fixed positive integer.
In this sequence, a(1), a(2) and a(4) are the largest terms of 2-sociable cycles of order 3 (or bicrowds), and a(3) is the larger term of a 3-sociable cycle of order 2 (or triamicable pair).
No other terms <= 10^12.

Crossrefs

Cf. A001065, A166069, A183013, A183021, A183022.

A183021 Smallest members of k-sociable cycles of order r, with k > 1 and r > 1.

Original entry on oeis.org

5974080, 12162100800, 90079209000, 555108915200
Offset: 1

Views

Author

William Rex Marshall, Jan 08 2011

Keywords

Comments

A k-sociable (or multisociable) cycle of order r consists of r distinct positive integers such that the sum of the aliquot divisors (or proper divisors) of each is equal to k times the next term in the cycle, where k (the multiplicity) is a fixed positive integer.
a(1), a(2) and a(4) are the smallest terms of 2-sociable cycles of order 3 (or bicrowds), and a(3) is the smaller term of a 3-sociable cycle of order 2 (or triamicable pair).
No other terms <= 10^12.

Crossrefs

Cf. A001065, A166069, A183016, A183020, A183022.

A183022 Multisociable numbers in cycles of order > 1 and multiplicity > 1.

Original entry on oeis.org

5974080, 7546128, 8132064, 12162100800, 13052008992, 14246719968, 90079209000, 97998179400, 555108915200, 574680043520, 582340505600
Offset: 1

Views

Author

William Rex Marshall, Jan 08 2011

Keywords

Comments

No other terms <= 10^12.

Crossrefs

Cf. A001065, A166069, A183019, A183020, A183021.

A244326 Numbers k such that floor(antisigma(k)/k) < floor(antisigma(k - 1)/(k - 1)).

Original entry on oeis.org

36, 48, 60, 72, 84, 90, 96, 108, 120, 132, 144, 156, 168, 180, 192, 210, 216, 240, 252, 264, 270, 280, 288, 300, 312, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 450, 456, 468, 480, 504, 510, 528, 540, 552, 560, 570, 576, 588, 600, 612, 624, 630, 648
Offset: 1

Views

Author

Jaroslav Krizek, Jun 25 2014

Keywords

Comments

Antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.
Numbers from A166069 (multiply perfect numbers k such that sigma(k)/k > 2) are members of this sequence.

Links

Crossrefs

Cf. A024816, A166069, A244324, A244325.

Programs

  • Magma
    [k: k in [2..10000] | Floor((((k*(k+1))div 2  - SumOfDivisors(k)) div k)) lt Floor((((k*(k-1))div 2  - SumOfDivisors(k-1)) div (k-1)))];
  • Mathematica
    With[{as=Table[Floor[Total[Complement[Range[2,n],Divisors[n]]/n]],{n,1000}]},Flatten[Position[Partition[as,2,1],?(First[#]>Last[#]&),{1},Heads->False]]]+1 (* _Harvey P. Dale, Sep 10 2014 *)

A340864 Numbers k such that both sigma_{-1}(k) > 2 and sigma_0(k)/sigma_{-1}(k) are integers.

Original entry on oeis.org

672, 30240, 32760, 2178540, 23569920, 45532800, 142990848, 459818240, 1379454720, 14182439040, 43861478400, 51001180160, 66433720320, 153003540480, 403031236608, 704575228896, 13661860101120, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904
Offset: 1

Views

Author

David Terr, Jan 24 2021

Keywords

Examples

			a(1) = 672 is the smallest number k that is both an Ore number and multiperfect such that sigma(k)/k > 2.

Links

Crossrefs

Intersection of A001599 and A166069.
Cf. A007691, A325025.

Programs

  • Mathematica
    Module[{a166069 = {120, 672, 30240, 32760, 523776, 2178540, 23569920, 45532800, 142990848, 459818240, 1379454720}, i, n, result = {}}, For[i = 1, i <= Length[a166069], i++, n = a166069[[i]]; If[Mod[DivisorSigma[0, n], DivisorSigma[-1, n]] == 0, AppendTo[result, n]]]; result]

Extensions

Name changed by and more terms from Jinyuan Wang, Feb 11 2021
Showing 1-8 of 8 results.

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

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