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

1,1

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

Links

Table of n, a(n) for n=1..59.

Example

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.

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 *)

Prog

(SageMath) 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())
(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

Crossrefs

Cf. A001414, A008472, A008683.

Cf. A059404, A323055.

Sequence in context: A359890 A059404 A323055 * A376250 A381719 A303946

Adjacent sequences: A383077 A383078 A383079 * A383081 A383082 A383083

Keyword

nonn

Author

Torlach Rush, Apr 15 2025

Status

approved