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

1,2

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001414, A007947, A359869.

Cf. A006881, A120944.

Sequence in context: A227299 A082092 A069116 * A063763 A115956 A066073

Adjacent sequences: A359867 A359868 A359869 * A359871 A359872 A359873

Keyword

nonn

Author

Johan Lindgren, Jan 16 2023

Status

approved