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 A334006 Triangle read by rows: T(n,k) = (the number of nonnegative bases m < n such that m^k == m (mod n))/(the number of nonnegative bases m < n such that -m^k == m (mod n)) for nonnegative k < n, n >= 1. 8
 1, 1, 1, 1, 3, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 7, 1, 3, 1, 3, 1, 1, 4, 1, 5, 1, 5, 1, 5, 1, 9, 1, 3, 1, 3, 1, 7, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 6, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 13, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 7, 1, 3, 1, 3 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS If the sum of proper divisors of q in row q <= q, then q are 1, 2, 3, 4, 5, 8, 16, 17, 32, 64, 128, 256, 257, ...(union of Fermat primes and powers of 2). LINKS Table of n, a(n) for n=1..97. EXAMPLE Triangle T(n,k) begins: n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ---+------------------------------------------------------ 1 | 1; 2 | 1, 1; 3 | 1, 3, 1; 4 | 1, 2, 1, 3; 5 | 1, 5, 1, 1, 1; 6 | 1, 3, 1, 3, 1, 3; 7 | 1, 7, 1, 3, 1, 3, 1; 8 | 1, 4, 1, 5, 1, 5, 1, 5; 9 | 1, 9, 1, 3, 1, 3, 1, 7, 1; 10 | 1, 5, 1, 1, 1, 5, 1, 1, 1, 5; 11 | 1, 11, 1, 3, 1, 3, 1, 3, 1, 3, 1; 12 | 1, 6, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9; 13 | 1, 13, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1; 14 | 1, 7, 1, 3, 1, 3, 1, 7, 1, 3, 1, 3, 1, 7; 15 | 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1, 3, 1, 15, 1; 16 | 1, 8, 1, 5, 1, 9, 1, 5, 1, 9, 1, 5, 1, 9, 1, 5; 17 | 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; ... For (n, k) = (7, 3), there are three nonnegative values of m < n such that m^3 == m (mod 7) (namely 0, 1, and 6) and one nonnegative value of m < n such that -m^3 == m (mod 7) (namely 0), so T(7,3) = 3/1 = 3. PROG (Magma) [[#[m: m in [0..n-1] | m^k mod n eq m]/#[m: m in [0..n-1] | -m^k mod n eq m]: k in [0..n-1]]: n in [1..17]]; (PARI) T(n, k) = sum(m=0, n-1, Mod(m, n)^k == m)/sum(m=0, n-1, -Mod(m, n)^k == m); matrix(7, 7, n, k, k--; if (k>=n, 0, T(n, k))) \\ to see the triangle \\ Michel Marcus, Apr 17 2020 CROSSREFS Cf. A000012, A000079, A002322, A010684, A019434, A023506, A026741, A182816, A333570, A337454, A337632, A337633, A337820, A337910. Sequence in context: A107296 A080847 A326406 * A364683 A270572 A095276 Adjacent sequences: A334003 A334004 A334005 * A334007 A334008 A334009 KEYWORD nonn,tabl AUTHOR Juri-Stepan Gerasimov, Apr 12 2020 EXTENSIONS Name corrected by Peter Kagey, Sep 12 2020 STATUS approved

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Last modified September 13 08:20 EDT 2024. Contains 375902 sequences. (Running on oeis4.)