A383080 Numbers k such that sopf(k) does not divide evenly sopfr(k).

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 198

Offset: 1

Author

Torlach Rush, Apr 15 2025

Keywords

• nonn

Comments

First differs from A059404 and A323055 at n = 59.

a(n) has a square factor, A008683(a(n)) = 0.

If p and q are distinct primes, p*q^k is in the sequence iff p + q does not divide k - 1. - Robert Israel, Apr 16 2025

Examples

			12 is a term because sopf(12)=5 does not evenly divide sopfr(12)=7.
18 is a term because sopf(18)=5 does not evenly divide sopfr(18)=8.
20 is a term because sopf(20)=7 does not evenly divide sopfr(20)=9.
		

Crossrefs

Cf. A001414, A008472, A008683.

Cf. A059404, A323055.

Programs

Maple

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[1]*t[2],t=F) mod add(t[1],t=F) <> 0
end proc:
select(filter, [$2..300]); # Robert Israel, Apr 16 2025

Mathematica

q[k_] := Module[{f = FactorInteger[k]}, !Divisible[Plus @@ Times @@@ f, Plus @@ f[[;; , 1]]]]; Select[Range[200], q] (* Amiram Eldar, Apr 16 2025 *)

PARI

isok(k) = if (k>1, my(f=factor(k)); sum(j=1, #f~, f[j,1]*f[j,2]) % sum(j=1, #f~, f[j,1])); \\ Michel Marcus, Apr 16 2025

Sage

def spf(k):
    fl = list(factor(k))
    sr = sum(p * e for p, e in fl)
    sd = sum(p for p, _ in fl)
    return sd, sr
def output(limit=198):
    results = []
    for k in range(2, limit + 1):
        sd, sr = spf(k)
        if 0 < sd and sr % sd != 0:
            results.append(k)
    return results
print(output())