A367419 a(n) = sqrt(A367418(n)).

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, 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, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Author

Amiram Eldar, Nov 17 2023

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot f(a(n)) at (x,y) = (n mod m, floor(n/m)) 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.

Crossrefs

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

Programs

Mathematica

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

PARI

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]-1), 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(sqrtint(b1), ", "))); }

Formula

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

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

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