cp's OEIS Frontend

A380487 Numbers such that the sum of prime factors without repetition divides the product of prime factors without repetition and each division yields a greater quotient.

%I A380487 #33 May 02 2025 04:21:49
%S A380487 2,30,70,105,231,627,805,1122,2730,3570,8778,9282,10626,15015,24738,
%T A380487 24882,31746,33495,33915,44330,45885,49335,51051,62985,72930,95095,
%U A380487 106590,132990,145145,156009,170170,222870,230945,274505,290598,329406,335478,418285,449995
%N A380487 Numbers such that the sum of prime factors without repetition divides the product of prime factors without repetition and each division yields a greater quotient.
%C A380487 Conjecture: There exists m, the smallest positive integer such that a(k)|a(k+m), for all k.
%H A380487 Robert Israel, <a href="/A380487/b380487.txt">Table of n, a(n) for n = 1..168</a>
%e A380487 2 is a term because sopf(2)|rad(2) = 2|2.
%e A380487 30 is a term because sopf(30)|rad(30) = 10|30.
%e A380487 70 is a term because sopf(70)|rad(70) = 14|70.
%p A380487 f:= proc(n,m) local q,S;
%p A380487    S:= numtheory:-factorset(n);
%p A380487    q:= convert(S,`*`)/convert(S,`+`);
%p A380487    if q::integer and q > m  then q else 0 fi;
%p A380487 end proc:
%p A380487 m:= 0: R:= NULL: count:= 0:
%p A380487 for n from 2 while count < 50 do
%p A380487   v:= f(n,m);
%p A380487   if v > 0 then
%p A380487     m:= v;  R:= R, n; count:= count+1;
%p A380487   fi;
%p A380487 od:
%p A380487 R; # _Robert Israel_, Apr 30 2025
%t A380487 s={};q=0;Do[pf=Times@@First/@FactorInteger[n];sf=Total[First/@FactorInteger[n]];If[Divisible[pf,sf]&&pf/sf>q,AppendTo[s,n];q=pf/sf],{n,2,449995}];s (* _James C. McMahon_, Apr 03 2025 *)
%o A380487 (Sage)
%o A380487 def sopf(n): return sum(set(prime_factors(n)))
%o A380487 def rad(n):
%o A380487     rad = 1
%o A380487     for p in set(prime_factors(n)): rad *= p
%o A380487     return rad
%o A380487 def output(limit=39):
%o A380487     results = []
%o A380487     n = 2
%o A380487     result = 0
%o A380487     while len(results) < limit:
%o A380487         sopf_n = sopf(n)
%o A380487         rad_n = rad(n)
%o A380487         if rad_n % sopf_n == 0 and result < rad_n / sopf_n:
%o A380487             results.append(n)
%o A380487             result = rad_n / sopf_n
%o A380487             print(n, end=', ')
%o A380487         n += 1
%o A380487     return results
%o A380487 output()
%o A380487 (PARI) lista(nn) = my(m=0, list=List()); for (n=2, nn, my(f=factor(n)[,1], q=factorback(f)/vecsum(f)); if ((denominator(q) == 1) && (q>m), listput(list, n); m=q);); Vec(list); \\ _Michel Marcus_, Mar 29 2025
%Y A380487 Cf. A007947, A008472.
%Y A380487 Subsequence of A086486.
%K A380487 nonn
%O A380487 1,1
%A A380487 _Torlach Rush_, Jan 24 2025