A304404 - OEIS

%I #12 Jul 12 2023 20:29:13

%S 1,1,1,1,1,6,1,1,1,10,1,12,1,14,15,1,1,18,1,20,21,22,1,24,1,26,1,28,1,

%T 900,1,1,33,34,35,36,1,38,39,40,1,1764,1,44,45,46,1,48,1,50,51,52,1,

%U 54,55,56,57,58,1,3600,1,62,63,1,65,4356,1,68,69,4900,1,72,1,74,75

%N If n = Product (p_j^k_j) then a(n) = Product (n/p_j^k_j).

%H Antti Karttunen, <a href="/A304404/b304404.txt">Table of n, a(n) for n = 1..65537</a>

%H Ilya Gutkovskiy, <a href="/A304404/a304404.jpg">Logarithmic scatter plot of a(n) up to n=30000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = n^(omega(n)-1), where omega() = A001221.

%F a(n) = A062509(n)/n.

%e a(60) = a(2^2*3*5) = (60/2^2) * (60/3) * (60/5) = 15 * 20 * 12 = 3600.

%t a[n_] := Times @@ (n/#[[1]]^#[[2]] & /@ FactorInteger[n]); Table[a[n], {n, 75}]

%t Table[n^(PrimeNu[n] - 1), {n, 75}]

%o (PARI) A304404(n) = (n^(omega(n)-1)); \\ _Antti Karttunen_, Aug 06 2018

%o (Python)

%o from sympy.ntheory.factor_ import primenu

%o def A304404(n): return int(n**(primenu(n)-1)) # _Chai Wah Wu_, Jul 12 2023

%Y Cf. A000961 (positions of ones), A001221, A003557, A007774 (fixed points), A028234, A028236, A051119, A062509, A066504, A069359, A072195, A141809, A205959, A284600, A290480.

%K nonn

%O 1,6

%A _Ilya Gutkovskiy_, May 12 2018