A367419 -id:A367419 - cp's OEIS frontend

A367407 a(n) = sqrt(A367406(n)).

1, 2, 3, 5, 6, 7, 4, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 12, 26, 9, 29, 30, 31, 8, 33, 34, 35, 37, 38, 39, 20, 41, 42, 43, 46, 47, 51, 53, 18, 55, 28, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 44, 89, 91, 93, 94, 95, 24, 97

Offset: 1

Amiram Eldar, Nov 17 2023

A permutation of the positive integers.

Cf. A064549, A268335, A367406, A367417, A367419.

Cf. A065463, A065465, A253905.

s[n_] := Sqrt[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(sqrtint(b1), ", ")));}

a(n) = sqrt(A064549(A268335(n))).
a(n) = sqrt(A268335(n)*A367417(n)).
a(n) = A268335(n)/A367419(n).
Sum_{k=1..n} a(k) = c * n^2 / 2, where c = (zeta(3)/(zeta(2)*d^2)) * Product_{p prime} (1 - 1/(p^2*(p+1))) = A253905 * A065465 / d^3 = 1.29812028442810841122..., and d = A065463 is the asymptotic density of the exponentially odd numbers (A268335).

A367418 The exponentially odd numbers (A268335) divided by their squarefree kernels (A007947).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 9, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Nov 17 2023

Analogous to A102631, with the exponentially odd numbers instead of the square numbers (A000290).

All the terms are square numbers.

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

Cf. A000290, A008833, A102631, A268335, A367406, A367417, A367419.

s[n_] := 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(b1, ", "))); }

a(n) = A003557(A268335(n)).
a(n) = A268335(n)/A367417(n).
a(n) = A367419(n)^2.
a(n) = A268335(n)^2/A367406(n).
a(n) = A008833(A268335(n)). - Amiram Eldar, Nov 30 2023

cp's OEIS Frontend

A367407 a(n) = sqrt(A367406(n)).

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