A367407 -id:A367407 - cp's OEIS frontend

A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

1, 4, 9, 25, 36, 49, 16, 100, 121, 169, 196, 225, 289, 361, 441, 484, 529, 144, 676, 81, 841, 900, 961, 64, 1089, 1156, 1225, 1369, 1444, 1521, 400, 1681, 1764, 1849, 2116, 2209, 2601, 2809, 324, 3025, 784, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761

Offset: 1

Amiram Eldar, Nov 17 2023

Analogous to A355038, with the exponentially odd numbers instead of the square numbers (A000290).

This sequence is a permutation of the square numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A007947, A053143, A064549, A065463, A268335, A355038, A367407, A367417, A367418.

s[n_] := n * Times @@ FactorInteger[n][[;;, 1]]; s /@ Select[Range[100], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

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(b1, ", ")));}

a(n) = A064549(A268335(n)).
a(n) = A268335(n)*A367417(n).
a(n) = A367407(n)^2.
a(n) = A268335(n)^2/A367418(n).
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = (Pi^2/(15*d^3)) * Product_{p prime} (1 - 1/(p^3*(p+1))) = 1.78385074227198915372..., and d = A065463 is the asymptotic density of the exponentially odd numbers.
a(n) = A053143(A268335(n)). - Amiram Eldar, Nov 30 2023

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

Original entry on oeis.org

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

Amiram Eldar, Nov 17 2023

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.

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

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

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), ", "))); }

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

cp's OEIS Frontend

A367406 The exponentially odd numbers (A268335) multiplied by their squarefree kernels (A007947).

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