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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, 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
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
KEYWORD
nonn
AUTHOR
F Cellarosi, Oct 24 2022
STATUS
approved