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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] = 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))<<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]

-- Reinhard Zumkeller, Mar 23 2012

CROSSREFS

Cf. A000010 (phi(n,0)), A058026 (phi(n,1)); 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).

Sequence in context: A246179 A166285 A336365 * A060116 A319068 A335423

Adjacent sequences:  A002469 A002470 A002471 * A002473 A002474 A002475

KEYWORD

nonn,nice,easy,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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