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

%I A385350 #22 Jun 28 2025 20:38:42
%S A385350 1,15,21,27,33,35,39,51,55,57,65,69,77,85,87,91,93,95,111,115,119,123,
%T A385350 125,129,133,141,143,145,155,159,161,177,183,185,187,201,203,205,209,
%U A385350 213,215,217,219,221,235,237,247,249,253,259,265,267,287,291,295,299
%N A385350 Numbers j such that the product of odd proper divisors of j is j.
%C A385350 Fixed points of A385349.
%C A385350 Odd terms in A007422.
%C A385350 Also 1 with odd numbers with exactly 4 divisors. - _David A. Corneth_, Jun 26 2025
%H A385350 Alois P. Heinz, <a href="/A385350/b385350.txt">Table of n, a(n) for n = 1..10000</a>
%H A385350 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePerfectNumber.html">Multiplicative Perfect Number</a>.
%p A385350 q:= n-> n=1 or n::odd and numtheory[tau](n)=4:
%p A385350 select(q, [$1..500])[];  # _Alois P. Heinz_, Jun 26 2025
%t A385350 A385349[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Select[Range[300], A385349[#] == # &]
%o A385350 (PARI) isok(k) = vecprod(select((x->((x%2)==1) && (x<k)), divisors(k))) == k; \\ _Michel Marcus_, Jun 26 2025
%o A385350 (PARI) is(n) = (n == 1) || (bitand(n, 1) && numdiv(n) == 4) \\ _David A. Corneth_, Jun 26 2025
%o A385350 (Python)
%o A385350 from math import isqrt
%o A385350 from sympy import primepi, integer_nthroot, primerange
%o A385350 def A385350(n):
%o A385350     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)))
%o A385350     m, k = n, f(n)
%o A385350     while m != k: m, k = k, f(k)
%o A385350     return m # _Chai Wah Wu_, Jun 27 2025
%Y A385350 Cf. A007422, A030513, A073582, A385349.
%K A385350 nonn,easy
%O A385350 1,2
%A A385350 _Ilya Gutkovskiy_, Jun 26 2025