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 A212737 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k lists the orders of degree-d irreducible polynomials over GF(prime(k)); listing order for each column: ascending d, ascending value. 12
 1, 1, 3, 1, 2, 7, 1, 2, 4, 5, 1, 2, 4, 8, 15, 1, 2, 3, 3, 13, 31, 1, 2, 5, 6, 6, 26, 9, 1, 2, 3, 10, 4, 8, 5, 21, 1, 2, 4, 4, 3, 8, 12, 10, 63, 1, 2, 3, 8, 6, 4, 12, 24, 16, 127, 1, 2, 11, 6, 16, 12, 6, 16, 31, 20, 17, 1, 2, 4, 22, 9, 3, 7, 8, 24, 62, 40, 51 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Alois P. Heinz, Antidiagonals n = 1..141, flattened Eric Weisstein's World of Mathematics, Irreducible Polynomial Eric Weisstein's World of Mathematics, Polynomial Order FORMULA Formulae for the column sequences are given in A059912, A212906, ... . EXAMPLE For k=1 the irreducible polynomials over GF(prime(1)) = GF(2) of degree 1-4 are: x, 1+x; 1+x+x^2; 1+x+x^3, 1+x^2+x^3; 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. The orders of these polynomials p (i.e., the smallest integer e for which p divides x^e+1) are 1; 3; 7; 5, 15. (Example: (1+x^3+x^4) * (1+x^3+x^4+x^6+x^8+x^9+x^10+x^11) == x^15+1 (mod 2)). Thus column k=1 begins: 1, 3, 7, 5, 15, ... . Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... 7, 4, 4, 3, 5, 3, 4, 3, 11, 4, ... 5, 8, 3, 6, 10, 4, 8, 6, 22, 7, ... 15, 13, 6, 4, 3, 6, 16, 9, 3, 14, ... 31, 26, 8, 8, 4, 12, 3, 18, 4, 28, ... 9, 5, 12, 12, 6, 7, 6, 4, 6, 3, ... 21, 10, 24, 16, 8, 8, 9, 5, 8, 5, ... 63, 16, 31, 24, 12, 14, 12, 8, 12, 6, ... 127, 20, 62, 48, 15, 21, 18, 10, 16, 8, ... MAPLE with(numtheory): M:= proc(n, i) M(n, i):= divisors(ithprime(i)^n-1) minus U(n-1, i) end: U:= proc(n, i) U(n, i):= `if`(n=0, {}, M(n, i) union U(n-1, i)) end: b:= proc(n, i) b(n, i):= sort([M(n, i)[]])[] end: A:= proc() local l; l:= proc() [] end; proc(n, k) local t; if nops(l(k))

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Last modified June 16 11:25 EDT 2024. Contains 373429 sequences. (Running on oeis4.)