A222009 (Product(primitive roots of p) - 1)/p, where p = prime(n) and n > 2.

1, 2, 61, 71, 684847, 8621, 4768743913, 192769238731, 31302497, 3624013907027, 3389284413733950439, 20347152500093, 73535243111830065216714893617, 579021662547635771462791245283, 38283945111344558723552263341142779661, 60296900399609972459, 271233083114844569997128597, 1382959355737627871079165208413804169

Offset: 3

Author

Jonathan Sondow, Feb 09 2013

Keywords

• nonn

Comments

Gauss proved that the product of the primitive roots of p is congruent to 1 modulo p, for all primes p except p = 3.

Examples

			The primitive roots of prime(4) = 7 are 3 and 5, and (3*5 - 1)/7 = 14/7 = 2, so a(4) = 2.
		

References

• C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 52.

Links

• Wikipedia, Primitive root

Crossrefs

Cf. A060749, A088145, A123475.

Programs

Mathematica

a[n_] := With[{p = Prime[n]}, Select[Range[p - 1], MultiplicativeOrder[#, p] == p - 1 &]]; Table[(Product[ a[n][[i]], {i, Length[a[n]]}] - 1)/Prime[n], {n, 3, 20}]

Formula

a(n) = (A123475(n) - 1)/A000040(n) for n > 2.