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A002472
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Number of pairs x,y such that y-x=2, (x,n)=1, (y,n)=1 and 1 <= x <= n.
(Formerly M0411 N0157)
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7
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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, 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, 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
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OFFSET
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1,4
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COMMENTS
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This is the function phi(n, 2) defined in Alder. - Michel Marcus, Nov 14 2017
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REFERENCES
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V. A. Golubev, Sur certaines fonctions multiplicatives et le problème des jumeaux. Mathesis 67 (1958), 11-20.
V. A. Golubev, Nombres de Mersenne et caractères du nombre 2. Mathesis 67 (1958), 257-262.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
a(n) = Sum_{k=1..n} [GCD(2*n-k,n) * GCD(k+2,n) = 1], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Sep 29 2021
Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/4) * Product_{p prime} (1 - 2/p^2) = (3/4) * A065474 = 0.2419755742... . - Amiram Eldar, Oct 23 2022
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EXAMPLE
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For n = 4, the condition gcd(x,4) = gcd(x+2,4) = 1 is satisfied by exactly two positive integers x not exceeding n, namely, by x = 1 and x = 3. Therefore a(4) = 2.
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MATHEMATICA
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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-François Alcover, Nov 22 2011 *)
(* Second program (5 times faster): *)
a[n_] := Sum[Boole[GCD[n, x] == 1 && GCD[n, x+2] == 1], {x, 1, n}];
f[p_, e_] := If[p == 2, p^(e-1), (p-2)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 22 2020 *)
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PROG
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(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
(PARI) a(n) = sum(x=1, n, (gcd(n, x) == 1) && (gcd(n, x+2) == 1)); \\ Michel Marcus, Nov 14 2017
(Haskell)
a002472 n = length [x | x <- [1..n], gcd n x == 1, gcd n (x + 2) == 1]
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CROSSREFS
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Similar generalizations of Euler's totient for prime k-tuples: this sequence (k=2), A319534 (k=3), A319516 (k=4), A321029 (k=5), A321030 (k=6).
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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