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 A212957 A(n,k) is the number of moduli m such that the multiplicative order of k mod m equals n; square array A(n,k), n>=1, k>=1, read by antidiagonals. 23
 0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 3, 2, 2, 2, 0, 2, 5, 4, 6, 1, 0, 4, 2, 3, 4, 4, 3, 0, 2, 6, 2, 12, 6, 10, 1, 0, 4, 4, 8, 4, 9, 16, 2, 4, 0, 3, 6, 2, 26, 4, 37, 6, 14, 2, 0, 4, 3, 12, 18, 4, 10, 3, 8, 4, 5, 0, 2, 12, 5, 14, 6, 42, 2, 28, 26, 16, 3, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Alois P. Heinz, Antidiagonals n = 1..60 Wikipedia, Multiplicative order FORMULA A(n,k) = |{m : multiplicative order of k mod m = n}|. A(n,k) = Sum_{d|n} mu(n/d)*tau(k^d-1), mu = A008683, tau = A000005. EXAMPLE A(4,3) = 6: 3^4 = 81 == 1 (mod m) for m in {5,10,16,20,40,80}. Square array A(n,k) begins:   0,  1,  2,  2,  3,  2,  4,  2, ...   0,  1,  2,  2,  5,  2,  6,  4, ...   0,  1,  2,  4,  3,  2,  8,  2, ...   0,  2,  6,  4, 12,  4, 26, 18, ...   0,  1,  4,  6,  9,  4,  4,  6, ...   0,  3, 10, 16, 37, 10, 42, 24, ...   0,  1,  2,  6,  3,  2, 12, 10, ...   0,  4, 14,  8, 28,  8, 48, 72, ... MAPLE with(numtheory): A:= (n, k)-> add(mobius(n/d)*tau(k^d-1), d=divisors(n)): seq(seq(A(n, 1+d-n), n=1..d), d=1..15); MATHEMATICA a[n_, k_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, k^d - 1], {d, Divisors[n]}]; a[1, 1] = 0; Table[ a[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 12 2012 *) CROSSREFS Columns k=1-10 give: A000004, A059499, A059885, A059886, A059887, A059888, A059889, A059890, A059891, A059892. Rows n=1-10 give: A000005, A059907, A059908, A059909, A059910, A059911, A218256, A218257, A218258, A218259. Main diagonal gives A252760. Cf. A000005, A008683. Sequence in context: A045832 A287528 A289281 * A035393 A068913 A128306 Adjacent sequences:  A212954 A212955 A212956 * A212958 A212959 A212960 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 01 2012 STATUS approved

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Last modified February 20 14:52 EST 2019. Contains 320327 sequences. (Running on oeis4.)