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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) 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS This is the function phi(n, 2) defined in Alder. - Michel Marcus, Nov 14 2017 REFERENCES 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). LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) Henry L. Alder, A Generalization of the Euler phi-Function, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 690-692. Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019. FORMULA 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 MATHEMATICA 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}]; Array[a, 81] (* Jean-François Alcover, Jun 19 2018, after Michel Marcus *) f[p_, e_] := If[p ==2, p^(e-1), (p-2)*p^(e-1)]; a = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 22 2020 *) PROG (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))<

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Last modified September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)