A080365 Composite numbers k whose smallest and largest prime factors are unitary prime factors.

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158

Offset: 1

Author

Labos Elemer, Feb 21 2003

Keywords

• nonn

Examples

			k=90 is not a prime; 90 = 2*3*3*5; extremal prime factors are 2 and 5; gcd(2, 90/2) = gcd(5, 90/5) = 1, so 2 and 5 are unitary prime divisors of 90, thus 90 is in the sequence.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..3000

Crossrefs

Cf. A034444, A056169, A020639, A006530, A080363, A080364.

Programs

GAP

D:=List(Filtered([2..160],i->not IsPrime(i)),Factors);;
a:=[];;  for i in D do if Gcd(i[1],Product(i)/i[1])=1 and Gcd(i[Length(i)],Product(i)/i[Length(i)])=1 then Add(a,Product(i)); fi; od; a;  # Muniru A Asiru, Jul 10 2018~

Mathematica

ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] mi[x_] := Part[Flatten[FactorInteger[x]], 1] k=0; Do[s=mi[n]; s1=ma[n]; If[Equal[GCD[s, n/s], 1]&&Equal[GCD[s1, n/s1], 1]&&!PrimeQ[n], Print[n]], {n, 2, 256}]

PARI

lista(nn) = {forcomposite(n=1, nn, my(f=factor(n)[,1], p = vecmin(f), q = vecmax(f)); if ((gcd(p, n/p) == 1) && (gcd(q, n/q) == 1), print1(n, ", ")););} \\ Michel Marcus, Jul 09 2018