cp's OEIS Frontend

A164722 Numbers whose sum of distinct prime factors is a square.

%I A164722 #21 Sep 08 2022 08:45:47
%S A164722 1,14,28,39,46,55,56,66,92,94,98,112,117,132,155,158,183,184,186,188,
%T A164722 196,198,203,224,255,264,275,290,291,295,299,316,323,334,351,354,368,
%U A164722 372,376,392,396,446,448,455,506,507,528,546,549,558,579,580,583,594
%N A164722 Numbers whose sum of distinct prime factors is a square.
%C A164722 This is to A008472 as A051448 is to A001414. It does seem that for any given k there should be a maximum n such that the sum of the prime factors of n = k^2, and a (perhaps different) maximum n such that the sum of distinct prime factors on n = k^2.
%C A164722 If k >= 3 and p = k^2 - 2 is prime (see A028870)  then 2 * p is the term. - _Marius A. Burtea_, Jun 12 2019
%H A164722 Marius A. Burtea, <a href="/A164722/b164722.txt">Table of n, a(n) for n = 1..14587</a> (terms up to 10^6)
%F A164722 {n such that A008472(n) = k^2 for k an integer}.
%F A164722 {n such that A008472(n) is in A000290}.
%e A164722 a(7) = 66 because 66 = 2 * 3 * 11 has sum of distinct prime factors 2 + 3 + 11 = 16 = 4^2. 8748 = 2^2 * 3^7 is the largest number whose prime factors (with multiplicity) add to 25 = 5^2, but it is not in this sequence because the sum of distinct prime factors of 8748 is 2 + 3 = 5, which is not a square.
%t A164722 Select[Range[600],IntegerQ[Sqrt[Total[Transpose[FactorInteger[#]] [[1]]]]]&] (* _Harvey P. Dale_, Mar 05 2014 *)
%o A164722 (PARI) isOK(n) = local(fac, i); fac = factor(n); issquare(sum(i=1, matsize(fac)[1], fac[i, 1])); \\ _Michel Marcus_, Mar 19 2013
%o A164722 (Magma) [n:n in [1..600]| IsPower(&+PrimeDivisors(n), 2)]; // _Marius A. Burtea_, Jun 12 2019
%Y A164722 Cf. A000290, A001414, A008472, A028870, A051448.
%K A164722 easy,nonn
%O A164722 1,2
%A A164722 _Jonathan Vos Post_, Aug 23 2009
%E A164722 More terms (including missing terms 56, 183, and 196) from _Jon E. Schoenfield_, May 27 2010