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A254309 Irregular triangular array read by rows: T(n,k) is the least positive primitive root of the n-th prime p=prime(n) raised to successive powers of k (mod p) where 1<=k<=p-1 and gcd(k,p-1)=1. 3

%I #27 May 04 2015 14:48:28

%S 1,2,2,3,3,5,2,8,7,6,2,6,11,7,3,10,5,11,14,7,12,6,2,13,14,15,3,10,5,

%T 10,20,17,11,21,19,15,7,14,2,8,3,19,18,14,27,21,26,10,11,15,3,17,13,

%U 24,22,12,11,21,2,32,17,13,15,18,35,5,20,24,22,19

%N Irregular triangular array read by rows: T(n,k) is the least positive primitive root of the n-th prime p=prime(n) raised to successive powers of k (mod p) where 1<=k<=p-1 and gcd(k,p-1)=1.

%C Each row is a complete set of incongruent primitive roots.

%C Each row is a permutation of the corresponding row in A060749.

%C Row lengths are A008330.

%C T(n,1) = A001918(n).

%H Alois P. Heinz, <a href="/A254309/b254309.txt">Rows n = 1..120, flattened</a>

%e 1;

%e 2;

%e 2, 3;

%e 3, 5;

%e 2, 8, 7, 6;

%e 2, 6, 11, 7;

%e 3, 10, 5, 11, 14, 7, 12, 6;

%e 2, 13, 14, 15, 3, 10;

%e 5, 10, 20, 17, 11, 21, 19, 15, 7, 14;

%e 2, 8, 3, 19, 18, 14, 27, 21, 26, 10, 11, 15;

%e Row 6 contains 2,6,11,7 because 13 is the 6th prime number. 2 is the least positive primitive root of 13. The integers relatively prime to 13-1=12 are {1,5,7,11}. So we have: 2^1==2, 2^5==6, 2^7==11, and 2^11==7 (mod 13).

%p with(numtheory):

%p T:= n-> (p-> seq(primroot(p)&^k mod p, k=select(

%p h-> igcd(h, p-1)=1, [$1..p-1])))(ithprime(n)):

%p seq(T(n), n=1..15); # _Alois P. Heinz_, May 03 2015

%t Table[nn = p;Table[Mod[PrimitiveRoot[nn]^k, nn], {k,Select[Range[nn - 1], CoprimeQ[#, nn - 1] &]}], {p,Prime[Range[12]]}] // Grid

%Y Cf. A001918, A008330, A060749.

%Y Last elements of rows give A255367.

%Y Row sums give A088144.

%K nonn,tabf

%O 1,2

%A _Geoffrey Critzer_, May 03 2015

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Last modified March 28 07:48 EDT 2024. Contains 371235 sequences. (Running on oeis4.)