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

Author

N. J. A. Sloane, Sep 08 2004

Keywords

• nonn
• nice

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

Crossrefs

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

Programs

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

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 *)

PARI

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

Formula

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