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A014567 Numbers n such that n and sigma(n) are relatively prime, where sigma(n) = sum of divisors of n, A000203. 35
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 36, 37, 39, 41, 43, 47, 49, 50, 53, 55, 57, 59, 61, 63, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 97, 98, 100, 101, 103, 107, 109, 111, 113, 115, 119, 121, 125, 127, 128, 129, 131, 133 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Related to "solitary numbers": n is solitary if there is no other integer m such that sigma(m)/m = sigma(n)/n.

It is easy to show that if n and sigma(n) are relatively prime then n is solitary. But the converse is not true; for example, 18, 45, 48 and 52 are solitary. Probably also 10, 14, 15, 20, 22 and many others are solitary, but I do not think that will ever be proved. - Dean Hickerson

From Daniel Forgues, Jun 23 2009: (Start)

Union of unit, primes and Duffinian numbers.

Duffinian numbers (A003624) are the composite numbers (including, among others, the proper prime powers) for which (n, sigma(n)) = 1. (End)

A009194(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2013

These numbers satisfy (denominator of sigma(n)/n) = n. - Michel Marcus, Oct 27 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

C. W. Anderson and D. Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84, 65-66, 1977.

Andrew Feist, Fun with the sigma(n) function, Missouri Journal of Mathematical Sciences 15:3 (2003), pp. 173-177.

P. A. Loomis, New families of solitary numbers, J. Algebra and Applications, 14 (No. 9, 2015), #1540004 (6 pages).

Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.

Eric Weisstein's World of Mathematics, Solitary Number.

FORMULA

a(n) << n log n. Can this be improved? - Charles R Greathouse IV, Feb 13 2013

a(n) >> n log log log n, see Luca. - Charles R Greathouse IV, Feb 17 2014

EXAMPLE

sigma(21) = 1 + 3 + 7 + 21 = 32 is relatively prime to 21, so 21 is in the sequence.

MATHEMATICA

lst={}; Do[d=DivisorSigma[1, n]; If[GCD[d, n]==1, AppendTo[lst, n]], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

Select[Range[150], CoprimeQ[#, DivisorSigma[1, #]]&] (* Harvey P. Dale, Jan 23 2015 *)

PROG

(PARI) is(n)=gcd(n, sigma(n))==1 \\ Charles R Greathouse IV, Feb 13 2013

(Haskell)

a014567 n = a014567_list !! (n-1)

a014567_list = filter ((== 1) . a009194) [1..]

-- Reinhard Zumkeller, Mar 23 2013

CROSSREFS

Cf. A003624.

Cf. A069059 (complement).

Sequence in context: A317923 A273130 A273200 * A324769 A326536 A322902

Adjacent sequences:  A014564 A014565 A014566 * A014568 A014569 A014570

KEYWORD

nonn,easy,nice

AUTHOR

Eric W. Weisstein

EXTENSIONS

More terms from Labos Elemer

STATUS

approved

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Last modified October 23 14:54 EDT 2019. Contains 328345 sequences. (Running on oeis4.)