A358015 a(n) = DedekindPsi(n*2^(-k))*2^(j-1) where k = valuation(n, 2) and j = k if 4 divides n and otherwise 0.

2, 2, 3, 2, 4, 4, 6, 3, 6, 8, 7, 4, 12, 8, 9, 6, 10, 12, 16, 6, 12, 16, 15, 7, 18, 16, 15, 12, 16, 16, 24, 9, 24, 24, 19, 10, 28, 24, 21, 16, 22, 24, 36, 12, 24, 32, 28, 15, 36, 28, 27, 18, 36, 32, 40, 15, 30, 48, 31, 16, 48, 32, 42, 24, 34, 36, 48, 24, 36, 48, 37, 19, 60

Offset

3,1

Comments

Reciprocals of the constants determining the tail of limiting distribution of quadratic Weyl sums with rational parameters.

Links

Table of n, a(n) for n=3..75.

Francesco Cellarosi and Tariq Osman, Heavy tailed and compactly supported distributions of quadratic Weyl sums with rational parameters, arXiv:2210.09838 [math.NT], 2022-2023.

Formula

Let DedekindPsi(m) = m * Product{p|m, p prime} (1 + 1/p).

Write n = m*2^k with odd m and k >= 0.

If k = 0 or k = 1, then const(n) = 2/DedekindPsi(m).

If k > 1, then const(n) = 1/(2^(k-1)*DedekindPsi(m)).

const(1) = const(2) = 1/2 and for all n >= 3, 1/const(n) is an integer.

a(n) = 1/const(n) for n >= 3.

From Peter Luschny, Oct 26 2022: (Start)

a(n) = (DedekindPsi(OddPart(n))/2) * [4 divides n ? EvenPart(n) : 1] = (A001615(A000265(n))/2) * [4 divides n ? A006519(n) : 1].

Define: k in S <=> k is a number that divides DedekindPsi(k) and its even part is greater than 2. Then sequence(k/12 for k in S) = sequence(a(k)/8 for k in S) = A003586. (End)

Sum_{k=1..n} a(k) ~ c * n^2, where c = 21/(8*Pi^2) = 0.265968... . - Amiram Eldar, Dec 09 2023

Example

a(3) = DedekindPsi(3)/2 = 4/2 = 2.
a(4) = 2^(2-1)*DedekindPsi(1) = 2*1 = 2.
a(5) = DedekindPsi(5)/2 = 6/2 = 3.
a(6) = DedekindPsi(3)/2 = 2.
a(16) = 2^(4-1)*DedekindPsi(1) = 8*1 = 8.
a(120) = 2^(3-1)*DedekindPsi(15) = 4*24 = 96.

Maple

alias(DedekindPsi = A001615):
A358015 := proc(n) local k, h; k := padic[ordp](n, 2); h := 2^k;
DedekindPsi(n/h)/2; if h > 2 then %*h else % fi end:
seq(A358015(n), n = 3..75); # Peter Luschny, Oct 25 2022

Mathematica

DedekindPsi[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}];
const[n_]:=If[OddQ[n] || (Mod[n, 4] == 2), 2/DedekindPsi[n/2^IntegerExponent[n, 2]], 1/(2^(IntegerExponent[n, 2]-1) DedekindPsi[n/2^IntegerExponent[n, 2]])]
a[n_]:=1/const[n]

Prog

(SageMath)
from sage.arith.misc import dedekind_psi
def A358015(n):
    k = valuation(n, 2)
    j = k if mod(n, 4) == 0 else 0
    return dedekind_psi(n*2^(-k))*2^(j-1)
print([A358015(n) for n in range(3, 76)]) # Peter Luschny, Oct 26 2022
(Python)
from math import prod
from sympy import primefactors
def A358015(n):
    s =(m:=n>>(k:=(~n & n-1).bit_length()))//prod(q:=primefactors(m))*prod(p+1 for p in q)
    return s>>1 if n&3 else s<<k-1 # Chai Wah Wu, Oct 27 2022

Crossrefs

Cf. A001615, A000265, A007814, A006519, A003586.

Sequence in context: A308318 A165419 A117660 * A184198 A238998 A355359

Adjacent sequences: A358012 A358013 A358014 * A358016 A358017 A358018

Keyword

nonn

Author

F Cellarosi, Oct 24 2022

Status

approved