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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.

%I A174842 #4 Feb 16 2025 08:33:12
%S A174842 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,
%T A174842 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,
%U A174842 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
%N 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.
%C A174842 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.
%H A174842 T. D. Noe, <a href="/A174842/b174842.txt">Rows n=1..500 of triangle, flattened</a>
%H A174842 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/ModuloMultiplicationGroup.html">MathWorld: Modulo Multiplication Group</a>
%F A174842 T(i,n) = phi(di), where di is the i-th divisor of prime(n)-1.
%e A174842 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.
%t A174842 Flatten[Table[EulerPhi[Divisors[p-1]], {p, Prime[Range[100]]}]]
%Y A174842 A008328 (tau(p-1)), A008330 (phi(p-1)), A174843 (divisors of p-1)
%K A174842 nonn,tabf
%O A174842 1,6
%A A174842 _T. D. Noe_, Mar 30 2010