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

A323652 Numbers m having at least one divisor d such that m divides sigma(d).

Original entry on oeis.org

1, 6, 12, 28, 56, 120, 360, 496, 672, 992, 2016, 8128, 16256, 30240, 32760, 60480, 65520, 120960, 131040, 523776, 1571328, 2178540, 4357080, 8714160, 23569920, 33550336, 45532800, 47139840, 67100672, 91065600, 94279680, 142990848, 182131200, 285981696
Offset: 1

Views

Author

Jaroslav Krizek, Jan 21 2019

Keywords

Comments

Generalization of multiperfect numbers (A007691).
Multiperfect numbers (A007691) are terms. If m is a k-multiperfect number and d divides k (for k > 1 and d > 1), then d*m is also a term.
Number 1379454720 is the smallest number with two divisors d with this property (459818240 and 1379454720). Another such number is 153003540480 with divisors 51001180160 and 153003540480. Is there a number with three divisors d with this property?
Supersequence of A081756.

Examples

			12 is a term because 6 divides 12 and simultaneously 12 divides sigma(6) = 12.

Crossrefs

Cf. A000203, A007691, A081756, A323653.

Programs

  • Magma
    [n: n in [1..10000] | #[d: d in Divisors(n) | SumOfDivisors(d) mod n eq 0] gt 0];
  • Mathematica
    Select[Range[530000],AnyTrue[DivisorSigma[1,Divisors[#]]/#,IntegerQ]&] (* The program generates the first 20 terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* Harvey P. Dale, Jan 17 2022 *)
  • PARI
    isok(n) = {fordiv(n, d, if (!(sigma(d) % n), return (1));); return (0);} \\ Michel Marcus, Jan 21 2019

A307741 Sum of divisors of the multiply-perfect numbers.

Original entry on oeis.org

1, 12, 56, 360, 992, 2016, 16256, 120960, 131040, 1571328, 8714160, 94279680, 67100672, 182131200, 571963392, 1379454720, 5517818880, 4428914688, 17179738112, 70912195200, 159991977600, 175445913600, 153003540480, 265734881280, 274877382656, 612014161920
Offset: 1

Views

Author

Jaroslav Krizek, Apr 26 2019

Keywords

Comments

When sorted, this is A081756. - N. J. A. Sloane, May 03 2019

Examples

			For n = 3; a(3) = sigma(A007691(3)) = sigma(28) = 56.

Links

Crossrefs

Cf. A000203, A007691, A054030, A088843 (tau(A007691)), A139256.
See also A081756.

Programs

  • Magma
    [SumOfDivisors(n): n in [1..1000000] | IsIntegral(SumOfDivisors(n)/n)]
  • PARI
    lista(nn) = {for (n=1, nn, my(s=sigma(n)); if (! (s % n), print1(s, ", ")););} \\ Michel Marcus, Apr 26 2019

Formula

a(n) = sigma(A007691(n)) = A000203(A007691(n)).
a(n) = A007691(n) * A054030(n).

A286917 Numbers k such that there is an anti-divisor d of k satisfying sigma(d) = k.

Original entry on oeis.org

3, 4, 13, 32, 40, 60, 121, 364, 1093, 3200, 3280, 9841, 15120, 16380, 29282, 29524, 88573, 91728, 264992, 265720, 797161, 2391484, 7174453, 21523360, 40098240, 64570081, 71495424, 78427440, 193690562, 193710244, 229909120, 581130733, 689727360, 1743392200, 5230176601
Offset: 1

Views

Author

Paolo P. Lava, May 16 2017

Keywords

Comments

As powers of 3 are in the sequence (larger than 1), the sequence is infinite. - David A. Corneth, Jul 20 2020

Examples

			Anti-divisors of 60 are 7, 8, 11, 17, 24, 40 and sigma(24) = 60.

Crossrefs

Cf. A000203, A014224, A066272, A081756, A130799.

Programs

  • Maple
    with(numtheory): P:= proc(q) local a,k,n; for n from 3 to q do a:=[];
for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=[op(a),k]; fi; od;
for k from 1 to nops(a) do if n=sigma(a[k]) then print(n); break; fi; od;
od; end: P(10^4); # Paolo P. Lava, May 16 2017
  • PARI
    isok(n) = {ad = select(t->n%t && tMichel Marcus, May 20 2017

Formula

sigma(3^m) is in the sequence, as is sigma(3^m*(3^(m + 1) - 2)) for prime 3^(m + 1) - 2. - David A. Corneth, Jul 20 2020

Extensions

More terms from Michel Marcus, May 20 2017
a(22)-a(26) from Jinyuan Wang, Jul 20 2020
a(27)-a(35) from David A. Corneth, Jul 20 2020
Showing 1-4 of 4 results.

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

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