A003624 Duffinian numbers: composite numbers k relatively prime to sigma(k). (Formerly M3324)

4, 8, 9, 16, 21, 25, 27, 32, 35, 36, 39, 49, 50, 55, 57, 63, 64, 65, 75, 77, 81, 85, 93, 98, 100, 111, 115, 119, 121, 125, 128, 129, 133, 143, 144, 155, 161, 169, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 225, 235, 237, 242, 243, 245, 247

Offset

1,1

Comments

All prime powers greater than 1 are in the sequence. No factorial number can be a term. - Arkadiusz Wesolowski, Feb 16 2014

Even terms are in A088827. Any term also in A005153 is either an even square or twice an even square not divisible by 3. - Jaycob Coleman, Jun 08 2014

All primes satisfy the second condition since gcd(p, p+1) = 1, thus making this sequence a proper subset of A014567. - Robert G. Wilson v, Oct 02 2014

References

T. Koshy, Elementary number theory with applications, Academic Press, 2002, p. 141, exerc. 6,7,8 and 9.

L. Richard Duffy, The Duffinian numbers, Journal of Recreational Mathematics 12 (1979), pp. 112-115.

Peter Heichelheim, There exist five Duffinian consecutive integers but not six, Journal of Recreational Mathematics 14 (1981-1982), pp. 25-28.

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 64.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

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

Rose Mary Zbiek, What can we say about the Duffinian numbers?, The Pentagon 42:2 (1983), pp. 99-109.

Formula

A009194(a(n)) * (1 - A010051(a(n))) = 1. - Reinhard Zumkeller, Mar 23 2013

a(n) >> n log log log n, see Luca. (Clearly excluding the primes only makes the n-th term larger.) - Charles R Greathouse IV, Feb 17 2014

Example

4 is in the sequence since it is not a prime, its divisors 1, 2, and 4 sum to 7, and gcd(7, 4) = 1.
21 is in the sequences since it is not a prime, and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21.

Mathematica

fQ[n_] := n != 1 && !PrimeQ[n] && GCD[n, DivisorSigma[1, n]] == 1; Select[ Range@ 280, fQ]

Prog

(PARI) is(n)=gcd(n, sigma(n))==1&&!isprime(n) \\ Charles R Greathouse IV, Feb 13 2013
(Haskell)
a003624 n = a003624_list !! (n-1)
a003624_list = filter ((== 1) . a009194) a002808_list
-- Reinhard Zumkeller, Mar 23 2013
(Python)
from math import gcd
from itertools import count, islice
from sympy import isprime, divisor_sigma
def A003624_gen(startvalue=2): # generator of terms
    return filter(lambda k:not isprime(k) and gcd(k, divisor_sigma(k))==1, count(max(startvalue, 2)))
A003624_list = list(islice(A003624_gen(), 30)) # Chai Wah Wu, Jul 06 2023

Crossrefs

Cf. A000203, A002808, A014567, A025475.

Sequence in context: A119315 A346256 A010390 * A280387 A243180 A100657

Adjacent sequences: A003621 A003622 A003623 * A003625 A003626 A003627

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein, Sep 19 1994

Status

approved