A389502 Primitive terms in A389501: powerful numbers whose coreful divisors have a noninteger arithmetic mean.

8, 16, 32, 125, 128, 216, 243, 256, 288, 392, 400, 432, 512, 864, 972, 1000, 1024, 1152, 1331, 1352, 1568, 1944, 2000, 2048, 2187, 2304, 2704, 2744, 2888, 3200, 3375, 3456, 3872, 3888, 4000, 4096, 4608, 4624, 4913, 5408, 5488, 5832, 6125, 6400, 6912, 7688, 7776

Offset

1,1

Comments

The coreful divisors of a number k are the divisors of k that have the same set of distinct prime factors as k (see A307958).

If k is a term, then k*m is a term in A389501 for any squarefree number m that is coprime to k, and is not divisible by den(k) = denominator(A005361(k)/A057723(k)) if den(k) is squarefree and coprime to k.

The asymptotic density of A389501 can be evaluated from the terms in this sequence (see the Comments section of A389501).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mathematica

pows[max_] := Union[Flatten[Table[i^2*j^3, {j, 1, Surd[max, 3]}, {i, 1, Sqrt[max/j^3]}]]];
f[p_, e_] := ((p^(e + 1) - 1)/(p - 1) - 1)/e;
q[n_] := !IntegerQ[Times @@ f @@@ FactorInteger[n]]; q[1] = False;
seq[max_] := Select[pows[max], q]; seq[8000]

Prog

(PARI) is1(k) = {my(f = factor(k), p, e); denominator(prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e+1)-1)/(p-1)-1)/e)) > 1; }
list(lim) = {my(s = List(), m); for(j = 1, sqrtnint(lim, 3), for(i = 1, sqrtint(lim\j^3), m = i^2*j^3; if(is1(m), listput(s, m)))); Set(s); }

Crossrefs

Intersection of A001694 and A389501.

Cf. A005117, A005361, A057723, A307958.

Sequence in context: A146541 A363014 A261976 * A291124 A239280 A043111

Adjacent sequences: A389499 A389500 A389501 * A389503 A389504 A389505

Keyword

nonn,easy

Author

Amiram Eldar, Oct 07 2025

Status

approved