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.

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

A325026 Multi-perfect numbers from A007691 that are not harmonic (A001599).

Original entry on oeis.org

120, 523776, 1476304896, 31998395520, 518666803200, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 170206605192656148480, 1161492388333469337600, 1802582780370364661760, 1940351499647188992000
Offset: 1

Views

Author

Jaroslav Krizek, Mar 24 2019

Keywords

Comments

Multi-perfect numbers m such that m*tau(m)/sigma(m) is not an integer, where tau(k) is the number of the divisors of k (A000005) and sigma(k) is the sum of the divisors of k (A000203).
Supersequence of A046987.
Complement of A325025 with respect to A007691.

Examples

			120 is a term because 120*tau(120)/sigma(120) = 120*16/360 = 16/3.

Links

Crossrefs

Cf. A000005, A000203, A001599, A007691, A325025.
Cf. A140798 (harmonic numbers that are not multi-perfect).

Programs

  • Magma
    [n: n in [1..1000000] | not IsIntegral((NumberOfDivisors(n)) * n / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]
  • PARI
    isok(n) = my(s=sigma(n)); !frac(s/n) && frac(n*numdiv(n)/s); \\ Michel Marcus, Mar 24 2019

A331724 Multiply-perfect numbers (A007691) that are arithmetic (A003601).

Original entry on oeis.org

1, 6, 672, 30240, 32760, 23569920, 45532800, 14182439040, 51001180160, 153003540480, 403031236608, 518666803200, 13661860101120, 740344994887680, 796928461056000, 212517062615531520, 87934476737668055040, 154345556085770649600, 170206605192656148480
Offset: 1

Views

Author

Jaroslav Krizek, Jan 25 2020

Keywords

Comments

Multiply-perfect numbers m such that values A(m) = sigma(m)/tau(m) = A000203(m)/A000005(m) are any integers.
Corresponding values of A(m): 1, 3, 84, 1260, 1365, 294624, 474300, 36933435, 318757376, 637514752, 1199497728, ...
Complement of A330533 with respect to A007691. Supersequence of A046985.
Has many terms in common with B = {multiply perfect numbers n divisible by bigomega(n)}: only {1, 45532800, 403031236608, 212517062615531520, ...} are in {a(n)} \ B, while {120, 523776, 2178540, ...} are in B \ {a(n)}. - M. F. Hasler, Jan 31 2020

Examples

			sigma(672)/tau(672) = 2016/24 = 84 (integers).

Links

Crossrefs

Cf. A000005, A000203, A003601, A007691, A046985, A330533.
Cf. A325025 (multiply-perfect numbers that are harmonic).

Programs

  • Magma
    [m: m in [1..10^7] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(SumOfDivisors(m) / m)];
  • Mathematica
    seqQ[n_] := And @@ (Divisible[DivisorSigma[1, n], #] & /@ {n, DivisorSigma[0, n]}); Select[Range[5*10^7], seqQ] (* Amiram Eldar, Jan 25 2020 *)
  • PARI
    is_A331724(n)={my(f=factor(n),s=sigma(f));!(s%n||s%numdiv(f))} \\ M. F. Hasler, Jan 31 2020

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 1-4 of 4 results.

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

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