A063994 a(n) = Product_{primes p dividing n } gcd(p-1, n-1).

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 4, 1, 16, 1, 18, 1, 4, 1, 22, 1, 4, 1, 2, 3, 28, 1, 30, 1, 4, 1, 4, 1, 36, 1, 4, 1, 40, 1, 42, 1, 8, 1, 46, 1, 6, 1, 4, 3, 52, 1, 4, 1, 4, 1, 58, 1, 60, 1, 4, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 4, 3, 4, 1, 78, 1, 2, 1, 82, 1, 16, 1, 4, 1, 88, 1, 36, 1

Offset: 1

Author

N. J. A. Sloane, Sep 18 2001

Keywords

• nonn
• easy
• nice

Comments

a(n) = number of bases b modulo n for which b^{n-1} == 1 (mod n).

a(A209211(n)) = 1. - Reinhard Zumkeller, Mar 02 2013

A049559(n) divides a(n) divides A000010(n). - Thomas Ordowski, Dec 14 2013

Note that a(n) = phi(n) iff n = 1 or n is prime or n is Carmichael number A002997. - Thomas Ordowski, Dec 17 2013

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
• R. Baillie and S. S. Wagstaff, Lucas pseudoprimes, Mathematics of Computation, 35 (1980), 1391-1417.
• P. Erdős and C. Pomerance, On the number of false witnesses for a composite number, Mathematics of Computation, 46 (1986), 259-279.
• Keith Gibson, NMBRTHRY posting, Sep 07, 2001.
• Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
• Carl Pomerance, NMBRTHRY posting, Jul 26, 2001.

Crossrefs

Cf. A002997, A027748.

Programs

Haskell

a063994 n = product $ map (gcd (n - 1) . subtract 1) $ a027748_row n
-- Reinhard Zumkeller, Mar 02 2013

Mathematica

f[n_] := Times @@ GCD[n - 1, First /@ FactorInteger@ n - 1]; f[1] = 1; Array[f, 92] (* Robert G. Wilson v, Aug 08 2011 *)

PARI

for (n=1, 1000, f=factor(n)~; a=1; for (i=1, length(f), a*=gcd(f[1, i] - 1, n - 1)); write("b063994.txt", n, " ", a) ) \\ Harry J. Smith, Sep 05 2009

PARI

a(n)=my(f=factor(n)[,1]);prod(i=1,#f,gcd(f[i]-1,n-1)) \\ Charles R Greathouse IV, Dec 10 2013

Python

def a(n):
    if n == 1: return 1
    return len([1 for witness in range(1,n) if pow(witness, n - 1, n) == 1])
[a(n) for n in range(1, 100)]

Formula

a(p^m) = p-1 and a(2p^m) = 1 for prime p and integer m > 0. - Thomas Ordowski, Dec 15 2013

a(n) = Sum_{k=1..n}(floor((k^(n-1)-1)/n)-floor((k^(n-1)-2)/n)). - Anthony Browne, May 11 2016

Extensions

More terms from Robert G. Wilson v, Sep 21 2001