A367419 - OEIS

%I #11 Nov 18 2023 10:55:50

%S 1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,3,1,1,1,4,1,1,1,1,1,1,2,1,1,1,

%T 1,1,1,1,3,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,

%U 4,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1

%N a(n) = sqrt(A367418(n)).

%H Amiram Eldar, <a href="/A367419/b367419.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A367419/a367419.png">Plot f(a(n)) at (x,y) = (n mod m, floor(n/m))</a> for m = 857 and n = 734449, where f is a color function such that 1 = gray, red indicates primes, gold composite prime powers, green squarefree composites, blue and purple numbers neither squarefree nor prime powers, but purple additionally represents squareful numbers that are not prime powers.

%F a(n) = sqrt(A003557(A268335(n))) = sqrt(A268335(n)/A367417(n)).

%F a(n) = A268335(n)/A367407(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/A065463 = 1.41956288050548591931... . - _Amiram Eldar_, Nov 17 2023

%t s[n_] := Sqrt[n / Times @@ FactorInteger[n][[;; , 1]]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

%o (PARI) b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]-1), 0)); }

%o lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(sqrtint(b1), ", "))); }

%Y Cf. A003557, A065463, A268335, A367407, A367417, A367418.

%K nonn,easy

%O 1,7

%A _Amiram Eldar_, Nov 17 2023