A083254 a(n) = 2*phi(n) - n.
1, 0, 1, 0, 3, -2, 5, 0, 3, -2, 9, -4, 11, -2, 1, 0, 15, -6, 17, -4, 3, -2, 21, -8, 15, -2, 9, -4, 27, -14, 29, 0, 7, -2, 13, -12, 35, -2, 9, -8, 39, -18, 41, -4, 3, -2, 45, -16, 35, -10, 13, -4, 51, -18, 25, -8, 15, -2, 57, -28, 59, -2, 9, 0, 31, -26, 65, -4, 19, -22, 69, -24, 71, -2, 5, -4, 43, -30, 77, -16, 27, -2, 81, -36, 43, -2, 25
Offset: 1
Examples
Case 1# - totient(x)-cototient[x] = 0 if x is a power of 2; Case 2# - totient(x)>cototient[x] gives odd primes and also A067800, (= A014076 except probably A036798); e.g. n = 33: a(33) = 2.20-33 = 7; n = p prime: a(p) = p-2; Case 3# - totient(x)<cototient[x] gives even numbers without powers of 2 and most probably A036798; e.g. n = 20: a(20) = -4; n = 105: a(105) = 2.48-105 = 96-105 = -9.
Links
- R. J. Mathar, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Maple
A083254 := proc(n) 2*numtheory[phi](n)-n ; end proc: # R. J. Mathar, Jan 13 2014
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Mathematica
Table[2*EulerPhi[w]-w, {w, 1, 1000}] -
PARI
a(n)=2*eulerphi(n)-n \\ Charles R Greathouse IV, Feb 21 2013
Formula
From Antti Karttunen, Dec 26 2017: (Start)
a(2n) = A297114(2n).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 6/Pi^2 - 1/2 = 0.107927... . - Amiram Eldar, Sep 07 2023
Comments