A359870 Numbers whose product of distinct prime factors is greater than the sum of its prime factors (with repetition).

1, 6, 10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 99, 102, 104, 105, 106, 110, 111, 114, 115, 116, 117

Offset: 1

Author

Johan Lindgren, Jan 16 2023

Keywords

• nonn

Comments

Numbers k where A007947(k) > A001414(k).

No term is prime since in that case the product of distinct prime factors and the sum of prime factors are equal.

Composite squarefree numbers (A120944) form a subsequence, so squarefree semiprimes (A006881) also. - Bernard Schott, Feb 22 2023

Examples

			45 = 3^2*5 is a term since its product of distinct prime factors 3 * 5 = 15 is greater than its sum of prime factors with multiplicity 3 + 3 + 5 = 11.
48 = 2^4*3 is not a term since its product of distinct prime factors 2 * 3 = 6 is less than its sum of prime factors with multiplicity 2 + 2 + 2 + 2 + 3 = 11.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A007947, A359869.

Cf. A006881, A120944.

Programs

Maple

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  mul(t[1],t=F) > add(t[1]*t[2],t=F);
end proc:
select(f, [$1..1000]); # Robert Israel, Feb 07 2023

Mathematica

q[n_] := Module[{f = FactorInteger[n]}, Times @@ f[[;; , 1]] > Plus @@ (f[[;; , 1]]*f[[;; , 2]])]; q[1] = True; Select[Range[120], q] (* Amiram Eldar, Jan 16 2023 *)

PARI

isok(n)={my(f=factor(n)); vecprod(f[,1]) > sum(i=1, #f~, f[i,1]*f[i,2])} \\ Andrew Howroyd, Jan 16 2023