A080365 - OEIS

%I #15 Jul 15 2018 12:23:05

%S 6,10,14,15,21,22,26,30,33,34,35,38,39,42,46,51,55,57,58,62,65,66,69,

%T 70,74,77,78,82,85,86,87,90,91,93,94,95,102,105,106,110,111,114,115,

%U 118,119,122,123,126,129,130,133,134,138,141,142,143,145,146,154,155,158

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

%H Muniru A Asiru, <a href="/A080365/b080365.txt">Table of n, a(n) for n = 1..3000</a>

%e 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.

%t 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}]

%o (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

%o (GAP) D:=List(Filtered([2..160],i->not IsPrime(i)),Factors);;

%o 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~

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

%K nonn

%O 1,1

%A _Labos Elemer_, Feb 21 2003