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A098006 (p-1)/2 - phi(p-1) as p runs through the odd primes. 8
0, 0, 1, 1, 2, 0, 3, 1, 2, 7, 6, 4, 9, 1, 2, 1, 14, 13, 11, 12, 15, 1, 4, 16, 10, 19, 1, 18, 8, 27, 17, 4, 25, 2, 35, 30, 27, 1, 2, 1, 42, 23, 32, 14, 39, 57, 39, 1, 42, 4, 23, 56, 25, 0, 1, 2, 63, 50, 44, 49, 2, 57, 35, 60, 2, 85, 72, 1, 62, 16, 1, 63, 66, 81, 1, 2, 78, 40, 76, 29, 114, 47 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,5

COMMENTS

In the Luca-Walsh paper it is shown that there are infinitely many numbers not in this sequence. See A098047.

a(n)=0 for Fermat primes (A019434). a(n)=1 for safe primes (A005385). a(n)=2 for A090866. The least prime p for which (p-1)/2-phi(p-1)=n or 0 if there is no such prime is given by A134765(n). Sequence A134854(k) gives the least prime for which a(n)=2^(k-1). For k not a power of 2, it can be shown that if k is in this sequence, then it appears for a prime p <= 1+k^2. - T. D. Noe, Nov 13 2007

REFERENCES

J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.

F. Luca and P. G. Walsh, On the number of nonquadratic residues which are not primitive roots, Colloq. Math., 100 (2004), 91-93.

LINKS

T. D. Noe, Table of n, a(n) for n = 2..10000

T. D. Noe, Finding primes p for which (p-1)/2 - phi(p-1) = k

FORMULA

a(n) = A005097(n-1) - A000010(A006093(n)); a(A159611(n)) = 0. - Reinhard Zumkeller, Mar 26 2013

MAPLE

A098006 := proc(n)

    local p;

    p := ithprime(n+1) ;

    (p-1)/2-numtheory[phi](p-1) ;

end proc:

seq(A098006(n), n=1..30) ; # R. J. Mathar, Jan 09 2017

MATHEMATICA

Table[(Prime[n] - 1)/2 - EulerPhi[Prime[n] - 1], {n, 2, 85}] (* Robert G. Wilson v, Sep 09 2004 *)

Table[(n-1)/2-EulerPhi[n-1], {n, Prime[Range[2, 100]]}] (* Harvey P. Dale, Oct 23 2016 *)

PROG

(PARI) forprime(p=3, 1e3, print1(p\2-eulerphi(p-1)", ")) \\ Charles R Greathouse IV, Feb 04 2013

(Haskell)

a098006 n = a005097 (n-1) - a000010 (a006093 n)

-- Reinhard Zumkeller, Mar 26 2013

(MAGMA) [(NthPrime(n)-1)/2 - EulerPhi(NthPrime(n)-1): n in [2..100]]; // Vincenzo Librandi, Jan 10 2017

CROSSREFS

Cf. A000010, A051953, A098047, A176095 (p runs through the odd numbers).

Sequence in context: A125943 A167565 A199470 * A336916 A082650 A054875

Adjacent sequences:  A098003 A098004 A098005 * A098007 A098008 A098009

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Sep 08 2004

STATUS

approved

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Last modified September 26 16:15 EDT 2021. Contains 347670 sequences. (Running on oeis4.)