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

1,1

Links

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

Example

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.

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}]

Prog

(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
(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~

Crossrefs

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

Sequence in context: A182853 A212168 A344585 * A000469 A120944 A327829

Adjacent sequences: A080362 A080363 A080364 * A080366 A080367 A080368

Keyword

nonn

Author

Labos Elemer, Feb 21 2003

Status

approved