A248020 Numbers which are coprime to the sum of their divisors, but are neither primes nor perfect powers.

21, 35, 39, 50, 55, 57, 63, 65, 75, 77, 85, 93, 98, 111, 115, 119, 129, 133, 143, 155, 161, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 235, 237, 242, 245, 247, 253, 259, 265, 275, 279, 291, 299, 301, 305, 309, 319, 323, 325, 327, 329, 333, 335, 338, 341

Offset: 1

Author

Robert G. Wilson v, Sep 29 2014

Keywords

• nonn
• easy

Comments

Intersection of A003624 and A106543. - Michel Marcus, Sep 30 2014

Duffinian numbers (A003624) which are not perfect powers (A001597). - Robert G. Wilson v, Oct 02 2014

Examples

			21 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21.
50 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 2, 5, 10, 25, and 50 sum to 93, which is coprime to 50.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Programs

Mathematica

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[ Range@ 350, !PrimeQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 && !perfectPowerQ[ #] &]
cpQ[n_]:=CoprimeQ[n,DivisorSigma[1,n]]&&!PrimeQ[n]&&GCD@@ FactorInteger[ n][[All,2]]<2; Select[Range[2,400],cpQ] (* Harvey P. Dale, Oct 05 2020 *)

PARI

forcomposite(n=1, 1e3, if(gcd(n, sigma(n))==1, if(!ispower(n), print1(n, ", ")))) \\ Felix Fröhlich, Oct 25 2014