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.

2, 30, 70, 105, 231, 627, 805, 1122, 2730, 3570, 8778, 9282, 10626, 15015, 24738, 24882, 31746, 33495, 33915, 44330, 45885, 49335, 51051, 62985, 72930, 95095, 106590, 132990, 145145, 156009, 170170, 222870, 230945, 274505, 290598, 329406, 335478, 418285, 449995

Offset: 1

Author

Torlach Rush, Jan 24 2025

Keywords

• nonn

Comments

Conjecture: There exists m, the smallest positive integer such that a(k)|a(k+m), for all k.

Examples

			2 is a term because sopf(2)|rad(2) = 2|2.
30 is a term because sopf(30)|rad(30) = 10|30.
70 is a term because sopf(70)|rad(70) = 14|70.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..168

Crossrefs

Cf. A007947, A008472.

Subsequence of A086486.

Programs

Maple

f:= proc(n,m) local q,S;
   S:= numtheory:-factorset(n);
   q:= convert(S,`*`)/convert(S,`+`);
   if q::integer and q > m  then q else 0 fi;
end proc:
m:= 0: R:= NULL: count:= 0:
for n from 2 while count < 50 do
  v:= f(n,m);
  if v > 0 then
    m:= v;  R:= R, n; count:= count+1;
  fi;
od:
R; # Robert Israel, Apr 30 2025

Mathematica

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 *)

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

Sage

def sopf(n): return sum(set(prime_factors(n)))
def rad(n):
    rad = 1
    for p in set(prime_factors(n)): rad *= p
    return rad
def output(limit=39):
    results = []
    n = 2
    result = 0
    while len(results) < limit:
        sopf_n = sopf(n)
        rad_n = rad(n)
        if rad_n % sopf_n == 0 and result < rad_n / sopf_n:
            results.append(n)
            result = rad_n / sopf_n
            print(n, end=', ')
        n += 1
    return results
output()