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Numbers whose product of distinct prime factors is greater than the sum of its prime factors (with repetition).
2

%I #45 Feb 24 2023 04:54:22

%S 1,6,10,14,15,20,21,22,26,28,30,33,34,35,38,39,42,44,45,46,51,52,55,

%T 56,57,58,60,62,63,65,66,68,69,70,74,75,76,77,78,82,84,85,86,87,88,90,

%U 91,92,93,94,95,99,102,104,105,106,110,111,114,115,116,117

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

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

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

%C Composite squarefree numbers (A120944) form a subsequence, so squarefree semiprimes (A006881) also. - _Bernard Schott_, Feb 22 2023

%H Robert Israel, <a href="/A359870/b359870.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%e 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.

%p filter:= proc(n) local F,t;

%p F:= ifactors(n)[2];

%p mul(t[1],t=F) > add(t[1]*t[2],t=F);

%p end proc:

%p select(f, [$1..1000]); # _Robert Israel_, Feb 07 2023

%t 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 *)

%o (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

%Y Cf. A001414, A007947, A359869.

%Y Cf. A006881, A120944.

%K nonn

%O 1,2

%A _Johan Lindgren_, Jan 16 2023