A385350 Numbers j such that the product of odd proper divisors of j is j.

1, 15, 21, 27, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299

Offset: 1

Author

Ilya Gutkovskiy, Jun 26 2025

Keywords

• nonn
• easy

Comments

Fixed points of A385349.

Odd terms in A007422.

Also 1 with odd numbers with exactly 4 divisors. - David A. Corneth, Jun 26 2025

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Crossrefs

Cf. A007422, A030513, A073582, A385349.

Programs

Maple

q:= n-> n=1 or n::odd and numtheory[tau](n)=4:
select(q, [$1..500])[];  # Alois P. Heinz, Jun 26 2025

Mathematica

A385349[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Select[Range[300], A385349[#] == # &]

PARI

isok(k) = vecprod(select((x->((x%2)==1) && (xMichel Marcus, Jun 26 2025

PARI

is(n) = (n == 1) || (bitand(n, 1) && numdiv(n) == 4) \\ David A. Corneth, Jun 26 2025

Python

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A385350(n):
    def f(x): return int(n-1+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jun 27 2025