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

A336702 Numbers whose abundancy index is a power of 2.

Original entry on oeis.org

1, 6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128
Offset: 1

Views

Author

Antti Karttunen, Aug 05 2020

Keywords

Comments

Apart from missing 2, this sequence gives all numbers k such that the binary expansion of A156552(k) is a prefix of that of A156552(sigma(k)), that is, for k > 1, numbers k for which sigma(k) is a descendant of k in A005940-tree. This follows because of the two transitions x -> A005843(x) (doubling) and x -> A003961(x) (prime shift) used to generate descendants in A005940-tree, using A003961 at any step of the process will ruin the chances of encountering sigma(k) anywhere further down that subtree.
Proof: Any left child in A005940 (i.e., A003961(k) for k) is larger than sigma(k), for any k > 2 [see A286385 for a proof], and A003961(n) > n for all n > 1. Thus, apart from A003961(2) = 3 = sigma(2), A003961^t(k) > sigma(k), where A003961^t means t-fold application of prime shift, here with t >= 1. On the other hand, sigma(2n) > sigma(n) for all n, thus taking first some doubling steps before a run of one or more prime shift steps will not rescue us, as neither will taking further doubling steps after a bout of prime shifts.
The first terms of A325637 not included in this sequence are 154345556085770649600 and 9186050031556349952000, as they have abundancy index 6.
From Antti Karttunen, Nov 29 2021: (Start)
Odd terms of this sequence are given by the intersection of A349169 and A349174.
A064989 applied to the odd terms of this sequence gives the fixed points of A326042, i.e., the positions of zeros in A348736, and a subset of the positions of ones in A348941.
Odd terms of this sequence form a subsequence of A348943, but should occur neither in A348748 nor in A348749.
(End)

Examples

			For 30240, sigma(30240) = 120960 = 4*30240, therefore, as sigma(k)/k = 2^2, a power of two, 30240 is present.

Links

Crossrefs

Cf. A000203, A003961, A005843, A005940, A156552, A163511, A209229, A286385.
Cf. A000396, A027687 (subsequences).
Subsequence of A007691, and after 1, also subsequence of A325637.
Union with {2} gives the positions of zeros in A347381.
Cf. also A326042, A347391, A347392, A347393, A347394, A348736, A348748, A348749, A348941, A348943, A349169, A349174.

Programs

  • PARI
    isA336702(n) = { my(r=sigma(n)/n); (1==denominator(r)&&!bitand(r, r-1)); }; \\ (Corrected) - Antti Karttunen, Aug 31 2021

A325636 a(n) = gcd(2n, sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 2, 1, 1, 2, 2, 4, 2, 4, 6, 1, 2, 3, 2, 2, 2, 4, 2, 12, 1, 2, 2, 56, 2, 12, 2, 1, 6, 2, 2, 1, 2, 4, 2, 10, 2, 12, 2, 4, 6, 4, 2, 4, 1, 1, 6, 2, 2, 12, 2, 8, 2, 2, 2, 24, 2, 4, 2, 1, 2, 12, 2, 2, 6, 4, 2, 3, 2, 2, 2, 4, 2, 12, 2, 2, 1, 2, 2, 56, 2, 4, 6, 4, 2, 18, 14, 8, 2, 4, 10, 12, 2, 1, 6, 1, 2, 12, 2, 2, 6
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Links

Crossrefs

Cf. A000203, A009194, A033879, A325385, A325635, A325637 (n such that a(n) = 2n).

Programs

  • PARI
    A325636(n) = gcd(n+n,sigma(n));

Formula

a(n) = gcd(2n, A000203(n)).
a(n) = gcd(2n, A033879(n)) = gcd(sigma(n), A033879(n)).

A325638 Numbers m such that sigma(m) can be obtained as the base-2 carryless product of 2m and some k.

Original entry on oeis.org

6, 28, 456, 496, 6552, 8128, 30240, 31452, 32760, 429240, 2178540, 7505976, 23569920, 33550336, 45532800, 142990848, 1379454720
Offset: 1

Views

Author

Antti Karttunen, May 21 2019

Keywords

Comments

Numbers m such that A000203(m) = A048720(2m, k) for some k.
Numbers m for which A091255(2m, sigma(m)) = 2m.
Conjecture: all terms are even. If this is true, then there are no odd perfect numbers. See also conjectures in A325639 and in A325808.

Links

Crossrefs

Cf. A000203, A091255, A325635, A325637, A325808.
Subsequence of A325639.
Cf. A000396 (a subsequence).

Programs

  • PARI
    A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A325635(n) = A091255sq(n+n, sigma(n));
isA325638(n) = ((n+n)==A325635(n));

Extensions

a(17) from Amiram Eldar, Jun 26 2024

A224832 Numbers k such that the sum of reciprocals of even divisors of k is an integer.

Original entry on oeis.org

12, 56, 992, 16256, 60480, 65520, 4357080, 47139840, 67100672, 91065600, 285981696, 2758909440, 17179738112, 87722956800, 132867440640, 274877382656, 306007080960, 806062473216, 1409150457792, 363485766938112, 12177456042320640, 29884246553283840, 40316371715063808
Offset: 1

Views

Author

Michel Lagneau, Jul 21 2013

Keywords

Comments

This sequence is a subsequence of A194771. The sequence A139256 (twice even perfect numbers) is a subsequence and the sum of the reciprocals of even divisors equals 1 (see the proof in this sequence). But, for the non-twice even perfect numbers of this sequence, for example a(5) = 60480, a(6) = 65520, a(7) = 4357080 so the sum of the reciprocals of even divisors equals 2.
Conjecture: if a(n) is a non-twice even perfect numbers, the sum of reciprocals of even divisors equals 2.

Examples

			12 is in the sequence because the divisors are {1, 2, 3, 4, 6, 12} and 1/2 + 1/4 + 1/6 + 1/12 = 1 is an integer.
67100672 is in the sequence because a(8)=A139256(5), the 5th Mersenne prime A000668(5) is 8191 = 2^13-1 and 8191*(8191+1) = 8191*8192 = 67100672.

Links

Crossrefs

Cf. A000668, A139256, A194771, A325637.

Programs

  • Maple
    with(numtheory):for n from 2 to 200000 do:x:=divisors(n):n1:=nops(x): s:=0:for i from 1 to n1 do: if irem(x[i],2)=0 then s:=s+1/x[i]:else fi:od: if s>0 and s=floor(s) then print(n):else fi:od:

Formula

a(n) = 2*A325637(n). - Amiram Eldar, Jun 26 2024

Extensions

a(17)-a(23) from Amiram Eldar, Jun 26 2024

A325654 Numbers m with a divisor d satisfying sigma(d) = 2*m.

Original entry on oeis.org

6, 28, 496, 8128, 60480, 65520, 4357080, 33550336, 47139840, 91065600, 285981696, 2758909440, 8589869056, 87722956800, 132867440640, 137438691328, 306007080960, 806062473216, 1409150457792, 363485766938112, 12177456042320640, 29884246553283840
Offset: 1

Views

Author

Jaroslav Krizek, May 12 2019

Keywords

Comments

Even perfect numbers from A000396 are terms.
Numbers of the form A007691(k)*A054030(k)/2 when A054030(k) is even.
Subsequence of A323652.
Numbers of the form sigma(A325637(k))/2. - Jinyuan Wang, Jun 09 2019

Examples

			60480 is a term because 30240 divides 60480 and sigma(30240) = 120960 = 2*60480.

Links

Crossrefs

Cf. A000203, A000396, A007691, A054030, A323652, A325637.

Programs

  • Magma
    [n: n in [1..100000] | #[d: d in Divisors(n) | SumOfDivisors(d) eq 2*n] gt 0];
  • PARI
    isok(n) = fordiv(n, d, if (sigma(d) == 2*n, return(1))); 0; \\ Michel Marcus, May 12 2019

Extensions

More terms from Jinyuan Wang, Jun 09 2019
Showing 1-6 of 6 results.

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

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