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

A066135 a(n) = least number m > 1 such that sigma_n(m) = k*m for some k.

Original entry on oeis.org

6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 228, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 84, 6, 10, 6, 34, 6, 10, 6, 194, 6, 10, 6
Offset: 1

Views

Author

Labos Elemer, Dec 06 2001

Keywords

Comments

a(n) <= 2p, where p = A002586(n) is the smallest prime factor of (1 + 2^n). (Proof. Since sigma_n(2p) = (1 + 2^n)(1 + p^n) and p is odd, 2p divides sigma_n(2p).) - Jonathan Sondow, Nov 23 2012

Links

Crossrefs

Cf. A000203, A001157, A001158, A001159, A002586, A007691, A046762-A046764, A055709-A055717.
Cf. A218860, A218861 (unique values and where they first occur).

Programs

  • Mathematica
    Table[m = 2; While[Mod[DivisorSigma[n, m], m] > 0, m++]; m, {n, 100}] (* T. D. Noe, Nov 23 2012 *)

Formula

Sum{d^n} = ka(n), d runs over the divisors of a(n), where k is an integer and a(n) is the smallest suitable number.

Extensions

Definition and formulas corrected by Jonathan Sondow, Nov 23 2012

A055715 Numbers k such that k | sigma_11(k).

Original entry on oeis.org

1, 6, 28, 120, 402, 496, 644, 672, 920, 1366, 1608, 1932, 2680, 2760, 3417, 3966, 4098, 4623, 4975, 5152, 6210, 6834, 8040, 8128, 8280, 9246, 9528, 9950, 12294, 13668, 15008, 15456, 15864, 16392, 18492, 19900, 24120, 24840, 25954, 27320, 27336, 29850, 30240, 32760
Offset: 1

Views

Author

Robert G. Wilson v, Jun 09 2000

Keywords

Comments

sigma_11(k) is the sum of the 11th powers of the divisors of k (A013959).

Links

Crossrefs

Cf. A013959.
Cf. A007691, A046762, A046763, A046764, A055709, A055710, A055711, A055712, A055713, A055714.

Programs

  • Mathematica
    Do[If[Mod[DivisorSigma[11, n], n]==0, Print[n]], {n, 1, 40000}]
  • PARI
    isok(k) = (sigma(k, 11) % k) == 0; \\ Michel Marcus, Nov 09 2019

Extensions

a(37)-a(40) corrected and more terms added by Amiram Eldar, Nov 09 2019

A076230 Numbers k such that sigma_2(k)/k and sigma_4(k)/k are integers.

Original entry on oeis.org

1, 84, 435708, 986076, 1441188, 6066396, 7407036, 16763292, 18735444, 78863148, 260855028, 318502548, 1340673516, 3391115364, 10829086632, 45076171140, 45582899544, 57648961188, 77314641768, 128293717860, 149098104540, 174975241896, 440359518060, 461191689504
Offset: 1

Views

Author

Labos Elemer, Oct 03 2002

Keywords

Examples

			For k = 6066396, sigma_2(k)/k = 9156979, sigma_4(k)/k = 241153415598179286943.

Links

Crossrefs

Intersection of A046762 and A046764.
Cf. A001157, A001159.

Programs

  • Magma
    [n: n in [1..10000000]|IsIntegral(DivisorSigma(2,n)/n) and IsIntegral(DivisorSigma(4,n)/n)]; // Marius A. Burtea, Aug 28 2019
  • Mathematica
    Select[Range[10^7], And @@ Divisible[DivisorSigma[{2, 4}, #], #] &] (* Amiram Eldar, Jan 20 2020 *)
  • PARI
    isok(n) = !(sigma(n, 2) % n) && !(sigma(n, 4) % n); \\ Michel Marcus, Aug 28 2019

Extensions

More terms from T. D. Noe, Apr 11 2006
a(11)-a(12) from Michel Marcus, Aug 28 2019
a(13)-a(18) from Amiram Eldar, Jan 20 2020
a(19)-a(24) from Giovanni Resta, Jan 21 2020
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.