login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 01:51 EST 2022. Contains 358672 sequences. (Running on oeis4.)