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 1-6 of 6 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

A061644 "Right perfect numbers": primes of the form 1 + a perfect number.

Original entry on oeis.org

7, 29, 33550337, 137438691329
Offset: 1

Views

Author

Labos Elemer, Jun 14 2001

Keywords

Comments

Readers of Rivera's web page (which I believe was indirectly based on this entry) later showed that there are no more cases among the first 39 perfect numbers. - N. J. A. Sloane, May 25 2004. The latest news is that there are no more cases among the first 44 perfect numbers. - M. F. Hasler, Jun 05 2008
So of the 44 known perfect numbers P=2^(p-1)*(2^p-1), P+1 is only prime for p=2,3,13 and 19.
If p is in the sequence then for each positive integer k, p^k is a solution to the equation sigma(phi(x)) = 2x-2. Proof: take t=2 in theorem related to the sequence A093034. [M. F. Hasler and Farideh Firoozbakht, Sep 09 2014]

Links

Crossrefs

Cf. A000396.
Analogous right and left multiple perfect numbers are in A093034, A094467.

Programs

  • Mathematica
    pn={6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216};lst={};Do[p=pn[[n]]+1;If[PrimeQ[p],AppendTo[lst,p]],{n,Length[pn]}];lst... and/or...PerfectNum[n_]:=Plus@@Divisors[n]/2;lst={};Do[p=PerfectNum[n];If[p==n&&PrimeQ[p+1],AppendTo[lst,p+1]],{n,10!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
  • PARI
    forprime(p=2,100,P=2^p-1;Q=P*(P+1)/2+1;if(isprime(P)&&isprime(Q),print1(Q,","))) \\ Edward Jiang, Sep 10 2014

Formula

P(p)*[P(p)+1]/2 + 1 is prime, where P(p) is a Mersenne prime. (corrected by Lekraj Beedassy, May 01 2009)

A171263 Primes of the form 1 + 4-multiperfect numbers.

Original entry on oeis.org

30241, 2178541, 23569921, 66433720321, 630532357710420079508428362350593, 1928622300236318049928258133164033, 651350717502447739281012140234441171379683329
Offset: 1

Views

Author

M. F. Hasler and Farideh Firoozbakht, Apr 15 2010

Keywords

Comments

A subsequence of A093034. If p is a term of this sequence then for each positive integer k, x=p^k is a solution for the equation sigma(phi(x))=4(x-1). See comment lines of the sequence A093034.

Examples

			p=30241 is prime and sigma(p-1)/(p-1)=4, so p is in the sequence.

Crossrefs

Cf. A027687, A093034, A171264, A171265.

A171264 Primes of the form 1 + 5-multiperfect numbers.

Original entry on oeis.org

30823866178561, 796928461056001, 1802582780370364661761, 2827987212986831882236723201, 68688966922031309945174465761834751373920047004215278394826366933532673
Offset: 1

Views

Author

M. F. Hasler and Farideh Firoozbakht, Apr 15 2010

Keywords

Comments

A subsequence of A093034. If p is a term of this sequence then for each positive integer k, x=p^k is a solution for the equation sigma(phi(x))=5(x-1). See comment lines of the sequence A093034.

Examples

			p=30823866178561 is prime and sigma(p-1)/(p-1)=5, so p is in the sequence.

Crossrefs

Cf. A027687, A093034, A171263, A171265.

A171265 Primes of the form 1 + 6-multiperfect numbers.

Original entry on oeis.org

9186050031556349952001, 837294585717161066223413745156096001, 102833200965458113792221682653890347008001, 8564360335859979576063023115658227351552001
Offset: 1

Views

Author

M. F. Hasler and Farideh Firoozbakht, Apr 15 2010

Keywords

Comments

Primes of the form 1+A046061(n). A subsequence of A093034. This sequence has only 8 terms. a(5)=1+A046061(76),a(6)=1+A046061(79),
a(7)=1+A046061(204) and a(8)=1+A046061(209).
If p is a term of this sequence then for each positive integer k, x=p^k is a solution for the equation sigma(phi(x))=6(x-1). See comment lines of the sequence A093034.

Examples

			p=9186050031556349952001 is prime and sigma(p-1)/(p-1)=6, so p is in the sequence.

Crossrefs

Cf. A046061, A093034, A171263, A171264.

A171266 Primes of the form 1 + triperfect numbers.

Original entry on oeis.org

673, 523777
Offset: 1

Views

Author

M. F. Hasler and Farideh Firoozbakht, Apr 15 2010

Keywords

Comments

A subsequence of A093034.
If p is a term of this sequence then for each positive integer k, x=p^k
is a solution for the equation sigma(phi(x))=3(x-1). See comment lines of
the sequence A093034.

Examples

			p=673 is prime and sigma(p-1)/(p-1)=3, so p is in the sequence.

Links

Crossrefs

Cf. A005820, A093034, A171263, A171264, A171265.
Showing 1-6 of 6 results.

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

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