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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 * A270572 A095276 A246457

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 April 14 05:27 EDT 2021. Contains 342944 sequences. (Running on oeis4.)