A098006 (p-1)/2 - phi(p-1) as p runs through the odd primes.

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

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