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.

A341044 Numbers k such that A318996(k) is prime.

Original entry on oeis.org

8, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 39, 42, 46, 57, 62, 65, 66, 69, 72, 74, 76, 80, 82, 87, 91, 92, 93, 94, 95, 100, 106, 111, 116, 119, 121, 122, 129, 133, 134, 145, 146, 159, 162, 166, 172, 176, 177, 183, 184, 190, 194, 202, 203, 206, 208, 213, 214, 215, 219, 232, 236, 237, 238, 240, 243
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Feb 03 2021

Keywords

Comments

Contains 2*p for p in A023201 and 3*p for p in A023203.

Examples

			a(3) = 10 is a term because A318996(10) = 11 is prime.

Links

Crossrefs

Cf. A023201, A023203, A318996.

Programs

  • Maple
    f:= proc(n) local s,D,t;
  D:= numtheory:-divisors(n);
  s:= convert(D,`+`);
  add(s mod t, t=D)
end proc:
select(t -> isprime(f(t)), [$1..1000]);
  • PARI
    f(n) = my(sn = sigma(n)); sumdiv(n, d, sn % d); \\ A318996
isok(k) = isprime(f(k)); \\ Michel Marcus, Feb 04 2021

A341045 Numbers k such that k divides A318996(k).

Original entry on oeis.org

1, 4, 6, 28, 45, 120, 496, 672, 6048, 8128, 14421, 30240, 32760, 523776, 2178540, 23569920, 26409026, 29270772, 30685402, 33550336, 45532800
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Feb 03 2021

Keywords

Examples

			a(2) = 4 is a term because A318996(4) = 4 is divisible by 4.
a(3) = 6 is a term because A318996(6) = 0 is divisible by 6.

Crossrefs

Contains A007691.
Cf. A318996.

Programs

  • Maple
    f:= proc(n) local s,D,t;
  D:= numtheory:-divisors(n);
  s:= convert(D,`+`);
  add(s mod t, t=D)
end proc:
select(t -> f(t) mod t = 0, [$1..20000]);
  • PARI
    f(n) = my(sn = sigma(n)); sumdiv(n, d, sn % d); \\ A318996
isok(k) = !(f(k) % k); \\ Michel Marcus, Feb 03 2021

A343904 Composite numbers k such that k is divisible by A318996(k).

Original entry on oeis.org

4, 45, 304, 464, 1870, 6048, 14421, 119072, 130304, 561824, 1736150, 8124416, 9519296, 16303104, 26409026
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, May 03 2021

Keywords

Examples

			a(4) = 464 is a member because it is composite and is divisible by A318996(k) = 16.

Crossrefs

Cf. A318996.

Programs

  • Maple
    filter:= proc(n) local D,s,d,v;
  if isprime(n) then return false fi;
D:= numtheory:-divisors(n);
s:= convert(D,`+`);
v:= add(s mod d, d = D);
  v > 0 and n mod v = 0
end proc:
select(filter, [$4..3*10^7]);

A349866 Fixed points of A318996.

Original entry on oeis.org

4, 45, 6048, 14421, 26409026, 29270772, 30685402
Offset: 1

Views

Author

Michel Marcus, Dec 03 2021

Keywords

Comments

Integers m such that Sum_{d|m} (sigma(m) mod d) = m.

Examples

			The sum of divisors of 4 is 7 with divisors 1, 2, 4; And (7 mod 1) + (7 mod 2) + (7 mod 4) = 0 + 1 + 3 = 4.

Links

Crossrefs

Cf. A000203, A318996.

Programs

  • PARI
    isok(m) = my(sn = sigma(m)); sumdiv(m, d, sn % d) == m;
  • Python
    from itertools import count, islice
from sympy import divisor_sigma, divisors
def A349866gen(): # generator of terms
    return filter(lambda m: sum(divisor_sigma(m) % d for d in divisors(m,generator=True)) == m, count(1))
A349866_list = list(islice(A349866gen(),4)) # Chai Wah Wu, Dec 03 2021

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
Showing 1-5 of 5 results.

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

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