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Revision History for A003165

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A003165

a(n) = floor(n/2) + 1 - d(n), where d(n) is the number of divisors of n.
(history; published version)

#60 by Nathaniel Johnston at Wed Apr 20 09:23:04 EDT 2022

STATUS

proposed

approved

#59 by Luca Onnis at Sat Apr 16 04:32:40 EDT 2022

STATUS

editing

proposed

#58 by Luca Onnis at Sat Apr 16 04:28:42 EDT 2022

COMMENTS

Conjecture: a(n) is the nullity of the (n-1) X (n-1) matrix M(n) with entries M(n)[i,j] = i*j mod n (matrices given by A352620). - Luca Onnis, Mar 27 2022

STATUS

proposed

editing

Discussion

Sat Apr 16

04:30

Luca Onnis: Okay, I manage to prove this .. it's very easy. The nullity is the order of the square matrix minus its rank, but a formula for the rank of those matrices has been already proven before.

04:32

Luca Onnis: As you can see from A088922 and A352620.

#57 by Luca Onnis at Sat Apr 16 04:19:07 EDT 2022

STATUS

editing

proposed

#56 by Luca Onnis at Sat Apr 16 04:18:58 EDT 2022

COMMENTS

Conjecture: a(n) is the algebraic multiplicity nullity of the eigenvalue 0 associated to the (n-1) X (n-1) matrix M(n) with entries M(n)[i,j] = i*j mod n (matrices given by A352620). - Luca Onnis, Mar 27 2022

STATUS

proposed

editing

#55 by Kevin Ryde at Thu Mar 31 20:25:12 EDT 2022

STATUS

editing

proposed

Discussion

Thu Mar 31

20:26

Kevin Ryde: Did we find a friendly linear algebra-ist to confirm or deny "nullity" ?

Fri Apr 01

02:01

Luca Onnis: Ok, thank you Kevin. No.. unfortunately I don't know any algebra-ist who can confirm that.

Sat Apr 02

18:57

Kevin Ryde: I found my textbook and had a little look around.  I think you could be like (if it's right!):

a(n+1) is the nullity of matrix M(n) from A352620, being the n X n matrix with entries M(n)[i,j] = i*j mod n+1.

Thu Apr 07

09:26

Luca Onnis: Okay Kevin thank you for your research and sorry for my late in answering you! You're right it's the nullity of those matrices.. Should I correct the conjecture and use the word "nullity"? Is it more precise? Let me know.

#54 by Kevin Ryde at Thu Mar 31 20:20:36 EDT 2022

MATHEMATICA

Table[Count[Eigenvalues[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}]], 0], {p, 2, 100}] (* Luca Onnis, Mar 27 2022 *)

STATUS

proposed

editing

Discussion

Thu Mar 31

20:24

Kevin Ryde: Ah, strict rule that program code cannot be based on conjecture.  (The existing looks better anyway - without actually trying!)

#53 by Luca Onnis at Thu Mar 31 12:42:40 EDT 2022

STATUS

editing

proposed

#52 by Luca Onnis at Thu Mar 31 12:42:11 EDT 2022

COMMENTS

Conjecture: a(n) is the algebraic multiplicity of the eigenvalue 0 associated to the (n-1) X (n-1) matrix M(n) with entries M(n)[i,j] = i*j mod n (matrices given by A352620). - Luca Onnis, Mar 27 2022

STATUS

proposed

editing

Discussion

Thu Mar 31

12:42

Luca Onnis: Conjecture associated to an empirical evidence.

#51 by Luca Onnis at Thu Mar 31 12:41:45 EDT 2022

STATUS

editing

proposed