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 1-5 of 5 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

A175200 Numbers k such that rad(k) divides sigma(k).

Original entry on oeis.org

1, 6, 24, 28, 40, 54, 96, 120, 135, 216, 224, 234, 270, 360, 384, 486, 496, 540, 588, 600, 640, 672, 864, 891, 936, 1000, 1080, 1350, 1372, 1521, 1536, 1638, 1782, 1792, 1920, 1944, 2016, 2160, 2176, 3000, 3240, 3375, 3402, 3456, 3564, 3724, 3744, 3780, 4320
Offset: 1

Views

Author

Michel Lagneau, Mar 03 2010

Keywords

Comments

rad(k) is the product of the distinct primes dividing k (A007947). sigma(k) is the sum of divisors of k (A000203). The odd numbers in this sequence (A336554) are rare: 1, 135, 891, 1521, 3375, 5733, 10935, 11907, 41067, 43875, ...
Also numbers k such that k divides sigma(k)^tau(k). - Arkadiusz Wesolowski, Nov 09 2013
This sequence is infinite. It contains an infinite number of even elements and an infinite number of odd ones. This is due to the fact that for every odd prime p and every prime q dividing p+1, p*q^r is prime-perfect when r = -1 + the multiplicative order of q modulo p. - Emmanuel Vantieghem, Oct 13 2014
For each term, it is possible to find an exponent k such that sigma(n)^k is divisible by n. A007691 (multi-perfect numbers) is a subsequence of terms that have k=1. A263928 is the subsequence of terms that have k=2. - Michel Marcus, Nov 03 2015
Pollack and Pomerance call these numbers "prime-abundant numbers". - Amiram Eldar, Jun 02 2020

Examples

			rad(6) = 6, sigma(6) = 12 = 6*2.
rad(24) = 6, sigma(24) = 60 = 6*10.
rad(43875) = 195, sigma(43875) = 87360 = 195*448.

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 827.

Links

Crossrefs

Cf. A000203, A007947, A027598, A069235, A105402, A173615, A336554 (odd terms).

Programs

  • Magma
    [n: n in [1..5000] | IsZero(DivisorSigma(1, n)^n mod n)];// Vincenzo Librandi, Aug 07 2018
  • Maple
    for n from 1 to 5000 do : p1:= ifactors(n)[2] :p2 :=mul(p1[i][1], i=1..nops(p1)): if irem(sigma(n),p2) =0 then print (n): else fi: od :
  • Mathematica
    Select[Range@5000, Divisible[DivisorSigma[1, #]^#, # ]&] (* Vincenzo Librandi, Aug 07 2018 *)
  • PARI
    isok(n) = {fs = Set(factor(sigma(n))[,1]); fn = Set(factor(n)[,1]); fn == setintersect(fn, fs);} \\ Michel Marcus, Nov 03 2015

A263983 Odd integers k such that sigma(k)^2 is divisible by k.

Original entry on oeis.org

1, 1521, 5733, 92781, 108927, 544635, 628849, 1886547, 2069613, 2723175, 2876211, 5659641, 16978923, 40847625, 45623655, 50936769, 60998353, 84418425, 84894615, 89162541, 173369889, 182995059, 191166885, 262840851, 401164731, 424473075, 426988471
Offset: 1

Views

Author

Michel Marcus, Oct 31 2015

Keywords

Comments

Odd terms of A263928.

Crossrefs

Cf. A000203, A017665, A017666, A263928.

Programs

  • Mathematica
    fQ[n_]:=IntegerQ[DivisorSigma[1,n]^2/n];Select[Range[1,1886547,2],fQ[#]&] (* Ivan N. Ianakiev, Nov 03 2015 *)
  • PARI
    lista(nn) = forstep(n=1, nn, 2, if (denominator(sigma(n)^2/n) == 1, print1(n, ", ")));

Extensions

a(21)-a(27) from Ivan N. Ianakiev, Oct 31 2015

A263940 Numbers such that the product of the sum of their anti-divisors and the sum of the reciprocals of their anti-divisors is an integer.

Original entry on oeis.org

3, 4, 6, 37, 96, 937, 2760, 393216
Offset: 1

Views

Author

Paolo P. Lava, Oct 30 2015

Keywords

Comments

A066466 is a subset of this sequence.
The sums are 1, 1, 1, 57, 1, 1457, 385, 1, ...

Examples

			Anti-divisors of 937 are 2, 3, 5, 15, 25, 75, 125, 375 and 625. Their sum is 1250 while the sum of their reciprocals is 1/2 + 1/3 + 1/5 + 1/15 + 1/25 + 1/75 + 1/125 + 1/375 + 1/625 = 1457/1250. Finally 1250 * 1457/1250 = 1457.

Crossrefs

Cf. A066272, A066466, A263928.

Programs

A344347 Numbers k such that sigma(k)^2 is divisible by k-1.

Original entry on oeis.org

2, 3, 5, 10, 33, 55, 82, 129, 136, 145, 261, 351, 385, 406, 442, 513, 649, 897, 1090, 2241, 4726, 5185, 8650, 13601, 17101, 17641, 18241, 26625, 26937, 29697, 29953, 32896, 34561, 35841, 38417, 44955, 46081, 46593, 51985, 63505, 65703, 66249, 84376, 93313, 97903
Offset: 1

Views

Author

Zdenek Cervenka, May 15 2021

Keywords

Examples

			For k=10, sigma(10)^2 / (10-1) = 18^2 / 9 = 324 / 9 = 36.

Crossrefs

Cf. A072861, A263928.

Programs

  • Mathematica
    Select[Range[2, 10^5], Divisible[DivisorSigma[1, #]^2, # - 1] &] (* Amiram Eldar, May 15 2021 *)
  • PARI
    list(nn) = for(n=2, nn, if (sigma(n)^2 % (n-1) == 0, print1(n, ", ")))
list(100000)
Showing 1-5 of 5 results.

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

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