A271547 -id:A271547 - cp's OEIS frontend

A070306 a(n) = 2*phi(n)/2^omega(n).

2, 1, 2, 2, 4, 1, 6, 4, 6, 2, 10, 2, 12, 3, 4, 8, 16, 3, 18, 4, 6, 5, 22, 4, 20, 6, 18, 6, 28, 2, 30, 16, 10, 8, 12, 6, 36, 9, 12, 8, 40, 3, 42, 10, 12, 11, 46, 8, 42, 10, 16, 12, 52, 9, 20, 12, 18, 14, 58, 4, 60, 15, 18, 32, 24, 5, 66, 16, 22, 6, 70, 12, 72, 18, 20, 18, 30, 6, 78, 16

Offset: 1

Benoit Cloitre, May 12 2002

Always an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Steven Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Cf. A000010, A001221, A271547.

a[n_] := EulerPhi[n]/2^(PrimeNu[n] - 1); Array[a, 100] (* Amiram Eldar, Apr 29 2022 *)

for(n=1,100,print1(2*eulerphi(n)/2^omega(n),","))

from sympy import totient as phi, primenu as omega
def a(n): return 2*phi(n)//2**omega(n)
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Apr 29 2022

Sum_{k=1..n} ~ c * n / sqrt(log(n)), where c = A271547/sqrt(Pi) (Finch and Sebah, 2006). - Amiram Eldar, Apr 29 2022

Offset changed to 1 and a(1) inserted by Amiram Eldar, Apr 29 2022

A269472 Decimal expansion of Product_{p prime} (1-(p^2+2)/(2(p^2+1)(p+1))) / sqrt(1-1/p), a constant related to the asymptotic average number of squares modulo n.

Original entry on oeis.org

1, 2, 5, 6, 9, 1, 3, 6, 1, 0, 2, 1, 0, 1, 8, 8, 5, 9, 5, 9, 4, 9, 2, 1, 1, 5, 7, 6, 9, 4, 6, 8, 6, 0, 8, 9, 4, 9, 4, 0, 4, 5, 9, 8, 8, 6, 8, 0, 7, 5, 0, 8, 7, 6, 7, 7, 9, 8, 5, 7, 1, 8, 1, 9, 3, 4, 7, 5, 1, 8, 2, 3, 8, 4, 5, 7, 4, 5, 4, 1, 4, 8, 7, 5, 5, 3, 9, 7, 5, 4, 8, 9, 7, 8, 6, 4, 9, 1, 1, 5, 7, 6, 4, 5, 0, 9, 9, 6

Offset: 1

Jean-François Alcover, Apr 13 2016

			1.2569136102101885959492115769468608949404598868075...

Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Cf. A046073, A105612, A271547.

digits = 104; m0 = 100; Clear[s]; s[m_] := s[m] = Sum[(1 + 2*(-1)^n - 4*(-1)^n*ChebyshevT[n, 1/4] + 4*Switch[Mod[n, 4], 2, -1, 3, 0, 0, 1, 1, 0])/(2*n) PrimeZetaP[n], {n, 2, m}] // N[#, digits]& // Exp; s[m0]; s[m = 2 m0]; While[RealDigits[s[m], 10, digits] != RealDigits[s[m/2], 10, digits], m = 2 m; Print[m]]; RealDigits[s[m]][[1]]
(* Second program: *)
$MaxExtraPrecision = 1000; Clear[f]; f[p_] := (1 - (p^2 + 2)/(2 (p^2 + 1) (p + 1)))/ Sqrt[1 - 1/p]; Do[c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 112]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 19 2020 *)

sqrt(prodeulerrat((1-(p^2+2)/(2*(p^2+1)*(p+1)))^2/(1-1/p))) \\ Amiram Eldar, May 29 2021

Formula in name and last digit corrected by Vaclav Kotesovec, Jun 19 2020

cp's OEIS Frontend

A070306 a(n) = 2*phi(n)/2^omega(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions