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 A002472 Number of pairs x,y such that y-x=2, (x,n)=1, (y,n)=1 and 1 <= x <= n. (Formerly M0411 N0157) 2

%I M0411 N0157

%S 1,1,1,2,3,1,5,4,3,3,9,2,11,5,3,8,15,3,17,6,5,9,21,4,15,11,9,10,27,3,

%T 29,16,9,15,15,6,35,17,11,12,39,5,41,18,9,21,45,8,35,15,15,22,51,9,27,

%U 20,17,27,57,6,59,29,15,32,33,9,65,30,21,15,69,12,71,35,15,34,45,11,77,24,27

%N Number of pairs x,y such that y-x=2, (x,n)=1, (y,n)=1 and 1 <= x <= n.

%C This is the function phi(n, 2) defined in Alder. - _Michel Marcus_, Nov 14 2017

%D V. A. Golubev, Nombres de Mersenne et caractÃ¨res du nombre 2. Mathesis 67 1958 257-262.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002472/b002472.txt">Table of n, a(n) for n = 1..1000</a>

%H Henry L. Alder, <a href="http://www.jstor.org/stable/2308710">A Generalization of the Euler phi-Function</a>, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 690-692.

%F Multiplicative with a(p^e) = p^(e-1) if p = 2; (p-2)*p^(e-1) if p > 2. - _David W. Wilson_, Aug 01 2001

%t a[n_] := If[ Head[ r=Reduce[ GCD[x, n] == 1 && GCD[x+2, n] == 1 && 1 <= x <= n, x, Integers]] === Or, Length[r], 1]; Table[a[n], {n, 1, 81}] (* _Jean-FranÃ§ois Alcover_, Nov 22 2011 *)

%o (PARI) a(n)=my(k=valuation(n,2),f=factor(n>>k));prod(i=1,#f[,1],(f[i,1]-2)*f[i,1]^(f[i,2]-1))<<max(0,k-1) \\ _Charles R Greathouse IV_, Nov 22 2011

%o (PARI) a(n) = sum(x=1, n, (gcd(n,x) == 1) && (gcd(n, x+2) == 1)); \\ _Michel Marcus_, Nov 14 2017

%o a002472 n = length [x | x <- [1..n], gcd n x == 1, gcd n (x + 2) == 1]

%o -- _Reinhard Zumkeller_, Mar 23 2012

%Y Cf. A000010 (phi(n,0)), A058026 (phi(n,1)).

%K nonn,nice,easy,mult

%O 1,4

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_

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Last modified February 20 01:26 EST 2018. Contains 299357 sequences. (Running on oeis4.)