A174842 Irregular triangle T(i,n) giving the number of elements of Zp having multiplicative order di, the i-th divisor of p-1, where p is the n-th prime.

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 2, 4, 1, 1, 2, 4, 8, 1, 1, 2, 2, 6, 6, 1, 1, 10, 10, 1, 1, 2, 6, 6, 12, 1, 1, 2, 4, 2, 4, 8, 8, 1, 1, 2, 2, 2, 6, 4, 6, 12, 1, 1, 2, 4, 4, 4, 8, 16, 1, 1, 2, 2, 6, 6, 12, 12, 1, 1, 22, 22, 1, 1, 2, 12, 12, 24, 1, 1, 28, 28, 1, 1, 2, 2, 4, 2, 4, 4, 8, 8

Offset: 1

Author

T. D. Noe, Mar 30 2010

Keywords

• nonn
• tabf

Comments

The divisors of p-1 are assumed to be in increasing order. The first row, for prime 2, has only one term. All other rows begin with two 1s and end with phi(p-1). There are tau(p-1), the number of divisors of p-1, terms in each row. The sum of the terms in each row is p-1. When p is a prime of the form 4k-1, then the last two terms in the row are equal. When p is a prime of the form 4k+1, then the last two terms in the row have a ratio of 2.

Examples

			For prime p=17, the 7th prime, the multiplicative order of the numbers 1 to p-1 is 1, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2. There is one 1, one 2, two 4's, four 8's, and eight 16's. Hence row 7 is 1, 1, 2, 4, 8.
		

Links

• T. D. Noe, Rows n=1..500 of triangle, flattened
• Eric W. Weisstein, MathWorld: Modulo Multiplication Group

Crossrefs

A008328 (tau(p-1)), A008330 (phi(p-1)), A174843 (divisors of p-1)

Programs

Mathematica

Flatten[Table[EulerPhi[Divisors[p-1]], {p, Prime[Range[100]]}]]

Formula

T(i,n) = phi(di), where di is the i-th divisor of prime(n)-1.