A340673 -id:A340673 - cp's OEIS frontend

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

Antti Karttunen and Peter Munn, Feb 01 2021

nonn

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.

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.

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

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).

A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900

Offset: 1

Antti Karttunen, Jan 31 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000079, A005117, A050376, A209229, A302777.
Positions of terms larger than 2 in A340673 (also in A340675), and of terms larger than 1 in A340676.
Subsequence of A072777 and of A340682.

Select[Range[10^4], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && u[[1]] == 2^IntegerExponent[u[[1]], 2] &] (* Amiram Eldar, Feb 08 2021 *)

A209229(n) = (n && !bitand(n,n-1));
isA340674(n) = { my(b,e); (((e=ispower(n,,&b))>0)&&issquarefree(b)&&A209229(e)); };

from math import isqrt
from sympy import mobius, integer_nthroot
def A340674(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,1<Chai Wah Wu, Jun 01 2025

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2^k)/zeta(2^(k+1))-1) = 0.6018231854... - Amiram Eldar, Feb 08 2021

A340676 If n is of the form s^(2^e), where s is a squarefree number, and e >= 0, then a(n) = 1+e, otherwise a(n) = 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 3, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 3, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 0, 1

Offset: 1

Antti Karttunen, Feb 01 2021

nonn

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

Cf. A005117, A104117, A225546, A267116, A297109, A340673, A340675.
Positions of zeros: {1} U A340681, of 1's: A005117 \ {1}, of 2's: A062503 \ {1}, of 3's: A113849.
Positions of nonzero terms: A340682, of terms > 1: A340674.

a[1] = 0; a[n_] := If[Length[(u = Union[FactorInteger[n][[;; , 2]]])] == 1 && u[[1]] == 2^(e = IntegerExponent[u[[1]], 2]), e + 1, 0]; Array[a, 100] (* Amiram Eldar, Feb 10 2021 *)

A001511(n) = 1+valuation(n,2);
A209229(n) = (n && !bitand(n,n-1));
A104117(n) = (A209229(n)*A001511(n));
A267116(n) = if(n>1, fold(bitor, factor(n)[, 2]), 0);
A340676(n) = if(1==n,0,A104117(A267116(n)));

a(n) = A297109(A225546(n)).

For n > 1, a(n) = A104117(A267116(n)). - Peter Munn, Feb 05 2021

cp's OEIS Frontend

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

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A340674 Numbers of the form s^(2^e), where s is a squarefree number, and e >= 1.

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A340676 If n is of the form s^(2^e), where s is a squarefree number, and e >= 0, then a(n) = 1+e, otherwise a(n) = 0.

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