A340675 Exponential of Mangoldt function conjugated by Tek's flip: a(n) = A225546(A014963(A225546(n))).

1, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 1, 2, 2, 2, 16, 2, 1, 2, 1, 2, 2, 2, 1, 4, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 16, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 4, 2, 2, 2, 1, 2

Offset: 1

Author

Antti Karttunen and Peter Munn, Feb 01 2021

Keywords

• nonn

Comments

Nonunit squarefree numbers take the value 2, other nonsquares take the value 1, and squares take the square of the value taken by their square root.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n
• Index entries for sequences related to prime signature

Crossrefs

Sequences used in a definition of this sequence: A014963, A048298, A225546, A267116, A297108, A340676.

Cf. A051903, A051144, A331591, A340673.

Positions of 1's: {1} U A340681, 2's: A005117 \ {1}, of 4's: A062503 \ {1}, of 16's: A113849.

Positions of terms > 1: A340682, of terms > 2: A340674.

Sequences used to express relationship between terms of this sequence: A003961, A331590.

Programs

PARI

A340675(n) = if(1==n,n,if(issquarefree(n), 2, if(!issquare(n), 1, A340675(sqrtint(n))^2)));

Formula

a(n) = A225546(A340673(n)) = A225546(A014963(A225546(n))).

a(n) = 2^A048298(A267116(n)).

If A340673(n) = 1, then a(n) = 1, otherwise a(n) = 2^A297108(A340673(n)).

If A340676(n) = 0, then a(n) = 1, otherwise a(n) = 2^(2^(A340676(n)-1)).

If n = s^(2^k), s squarefree >= 2, k >= 0, then a(n) = 2^(2^k), otherwise a(n) = 1.

For n, k > 1, if a(n) = a(k) then a(A331590(n, k)) = a(n), otherwise a(A331590(n, k)) = 1.

a(n^2) = a(n)^2.

a(A003961(n)) = a(n).

a(A051144(n)) = 1.

a(n) = 1 if and only if A331591(n) <> 1, otherwise a(n) = 2^A051903(n).