Revision History for A329534
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#56 by Charles R Greathouse IV at Thu Sep 08 08:46:24 EDT 2022
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| PROG
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(MAGMAMagma) [[k: k in [1..n] | k^2 mod n eq k]: n in [1..38]];
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Discussion
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Thu Sep 08
| 08:46
| OEIS Server: https://oeis.org/edit/global/2944
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#55 by N. J. A. Sloane at Sun Feb 23 16:16:42 EST 2020
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#54 by Michael De Vlieger at Wed Feb 05 18:24:32 EST 2020
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#53 by Michael De Vlieger at Wed Feb 05 18:23:50 EST 2020
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| MATHEMATICA
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Table[Select[Range@ n, Mod[-n + #^2, + # (# - 1), n] == # &], {] == 0 &], {n, 3925}] // Flatten (* Michael De Vlieger, Nov 18 2019 *)
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| STATUS
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proposed
editing
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Discussion
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Wed Feb 05
| 18:24
| Michael De Vlieger: Fixed Mathematica per Wolfdieter's pink box comment immediately preceding this one.
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#52 by Wolfdieter Lang at Wed Feb 05 13:50:17 EST 2020
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#51 by Wolfdieter Lang at Wed Feb 05 13:29:29 EST 2020
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| NAME
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Irregular triangle read by rows: for wheren >= 1 row n > 1 lists the positive k <= n - 1 from [1, 2, ... , n] such that A002378(k-1) = (k == --1)*k^2 == 0 (mod n).
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| DATA
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1, 1, 2, 1, 3, 1, 4, 2, 31, 5, 6, 71, 83, 4, 56, 91, 107, 31, 8, 11, 12, 61, 79, 131, 5, 6, 910, 141, 1511, 161, 84, 9, 1712, 181, 413, 151, 197, 68, 14, 20, 10, 11, 211, 226, 810, 15, 23, 241, 1216, 131, 2517, 261, 79, 2010, 2718, 281, 519, 91, 145, 1516, 20, 24, 29, 301, 317, 1115, 21, 32, 16, 17, 33, 14, 201, 3411, 812, 2722, 351, 3623, 181, 199, 3716, 1224, 261, 3825
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| OFFSET
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1,23
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| COMMENTS
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n-th row length gives 1 for A309307(n) = A000225( = 1, and 2^A001221(n) for n)) = >= 2 , that is A034444(n) - 1). [Proof: Unique lifting theorem (e.g., Apostol, 5.30 (a), p.121) for this congruence, and only two solutions 1 and p for primes p. See also the Yuval Dekel, Sep 21 2003, comment in A034444. - _Wolfdieter Lang_, Feb 05 2020]
A000961(n)-th row length is a nonprime number of the form 2^x - 1.
A024619(n)-th row length is a Mersenne prime.
A006881(n)-th row length = 3.
n-th row sum gives (n-1)*A007875(n) = (n-1)*A000010(2^A001221(n)).
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| REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
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| EXAMPLE
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n\k 1 2 3 ... 4 ...
1: 1:
2: : 1 2
3: 2
4: 3
5: 4
6: 2 3 5
7: 6
8: 7
9: 8
10: 4 5 9
3: 1 3
4: 1 4
5: 1 5
6: 1 3 4 6
7: 1 7
8: 1 8
119: 1 9
12: 3 8 11
13: 12
14: 6 7 13
15: 5 9 14.
10: 1 5 6 10
11: 1 11
12: 1 4 9 12
13: 1 13
14: 1 7 8 14
15: 1 6 10 15
16: 1 16
17: 1 17
18: 1 9 10 18
19: 1 19
20: 1 5 16 20
...
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| MATHEMATICA
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Table[Select[Range@ n, Mod[[-n - #^ + #^2, n] == # &], {n, 39}] // Flatten (* Michael De Vlieger, Nov 18 2019 *)
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| PROG
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(MAGMA) [[k: k in [1..n-1] | -] | k^2 mod n eq k]: n in [1..38]];
(PARI) row(n) = select(x->(Mod(x, n) == -) == Mod(x, n)^2), [1..n-1]); \\ Michel Marcus, Nov 19 2019
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| CROSSREFS
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Cf. A000010, A000225, A000688, A000961, A001221, A006881, A006530, A007875, A020639, A024619, A034444, A077610, A135972, A309307.
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| KEYWORD
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nonn,easy,tabf
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| EXTENSIONS
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Edited by Wolfdieter Lang, Feb 05 2020
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| STATUS
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proposed
editing
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Discussion
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Wed Feb 05
| 13:35
| Wolfdieter Lang: Since there came no answer from Juri-Stepan Gerasimov I edited this sequence by changing k -> k-1, and the range of k is now [1...n] (not [1..n-1]. I used [1..n] insted of [0..(n-1)] in order to compare this with the ireegular triangle A077610.// I changed the MAGMA nad PARI program accordingly ( is it ok?) , and wrote to Michael De Vlieger to change the Mma program.
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#50 by Michel Marcus at Sun Dec 08 12:59:25 EST 2019
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Discussion
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Sat Jan 04
| 21:53
| Sean A. Irvine: This doesn't look like a triangle to me, not even an irregular one.
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Thu Jan 09
| 13:26
| Wolfdieter Lang: To Juri-Stepan Gerasimov: with some small redefinition this seems to become the irregular triangle A077610 (n, m) ( n-th row lists unitary divisors of n) with offset n >= 1, and row length A034444(n). If true I suggest to recycle this proposal. / /
The redefinition is as follows: Do not consider A002378(k) = k^2 + k == 0 (mod n), with representative k (from {0, 1 ,...,n-1}, but use A002378(k-1) = (k-1)*k == 0 (mod n), n >= 1 and k from {1, 2, ... ,n}. Then this becomes the irregular triangle T(n, m) = A077610 (n, m) , with m = 1, 2, ..., A034444(n).
Do you agree?
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| 22:58
| Wolfdieter Lang: To Juri-Stepan Gerasimov: Please erase in my comment the 'seems to', and replace 'become' by 'becomes', because gcd(k, (k-1)*k/k) = gcd(k, k-1) = gcd( k, (k-1) -1*k) = gcd(k, -1) = gcd(k, 1) = 1. Therefore, this entry should be recycled, and one could instead give a comment in A077610 like "Row n gives the integers k with 1 <= k <= n and A002378(k-1) = K^2 -k == 0 (mod n), for n >= 1."
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| 23:00
| Wolfdieter Lang: It should have been: A002378(k-1) = k^2 -k == 0 (mod n). Sorry.
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Fri Jan 10
| 00:46
| Wolfdieter Lang: To To Juri-Stepan Gerasimov: Please forget all my comments above. I am very, very sorry. I was thinking about a different irregular triangle, namely [[1], [1, 2], [1, 3], [1, 4], [1, 5], [1, 3, 4, 6], [1, 7], [1, 8], [1, 9], [1, 5, 6, 10] , ...] which you could consider instead of the one you gave. But this is NOT A077610 (compare row n=6) . This seems not to be in OEIS. The definition is: Row n gives the integers k with 1 <= k <= n and A002378(k-1) = k^2 - k == 0 (mod n), for n >= 1. The row length seems to be A034444(n). What do you think about giving this triangle instead of the one you gave?
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Mon Jan 13
| 14:55
| Wolfdieter Lang: To the editors: I have writen an e-mail to Juri-Stepan Gerasimov (Jan 10).
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Thu Jan 30
| 16:46
| Wolfdieter Lang: Today I have repeated the mail .
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#49 by Michel Marcus at Sun Dec 08 12:59:19 EST 2019
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| COMMENTS
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A024619(n)-th row length is a Mersenne prime.
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| STATUS
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proposed
editing
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#48 by Juri-Stepan Gerasimov at Sun Dec 08 12:34:40 EST 2019
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#47 by Juri-Stepan Gerasimov at Sun Dec 08 12:32:23 EST 2019
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| COMMENTS
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A000961(n)-th row length <= is a nonprime number of the form 2^x - 1.
A024619(n)-th row length is Mersenne prime.
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| CROSSREFS
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Cf. A000010, A000225, A000688, A000961, A001221, A006881, A006530, A007875, A020639, A024619, A034444, A135972, A309307.
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| STATUS
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proposed
editing
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