A047209 -id:A047209 - cp's OEIS frontend

A323674 Square array, read by antidiagonals, of the positive integers 6cd +-c +-d = (6c +- 1)d +- c. Alternate rows (or columns) are numbers that differ by c from multiples of 6c - 1 or 6c + 1.

4, 6, 6, 9, 8, 9, 11, 13, 13, 11, 14, 15, 20, 15, 14, 16, 20, 24, 24, 20, 16, 19, 22, 31, 28, 31, 22, 19, 21, 27, 35, 37, 37, 35, 27, 21, 24, 29, 42, 41, 48, 41, 42, 29, 24, 26, 34, 46, 50, 54, 54, 50, 46, 34, 26, 29, 36, 53, 54, 65, 60, 65, 54, 53, 36, 29, 31, 41, 57, 63, 71, 73, 73, 71, 63, 57, 41, 31

Offset: 1

Sally Myers Moite, Jan 23 2019

This sequence without duplicates is A067611, which is the complement of A002822, the positive integers x for which 6x - 1 and 6x + 1 are twin primes.

			Square array begins:
   4,   6,   9,  11,  14,  16,  19,  21,  24,  26, ...
   6,   8,  13,  15,  20,  22,  27,  29,  34,  36, ...
   9,  13,  20,  24,  31,  35,  42,  46,  53,  57, ...
  11,  15,  24,  28,  37,  41,  50,  54,  63,  67, ...
  14,  20,  31,  37,  48,  54,  65,  71,  82,  88, ...
  16,  22,  35,  41,  54,  60,  73,  79,  92,  98, ...
  19,  27,  42,  50,  65,  73,  88,  96, 111, 119, ...
  21,  29,  46,  54,  71,  79,  96, 104, 121, 129, ...
  24,  34,  53,  63,  82,  92, 111, 121, 140, 150, ...
  26,  36,  57,  67,  88,  98, 119, 129, 150, 160, ...
  ...
Note that, for example, the third row (or column) contains numbers that differ by 2 from multiples of 11 = 6*2 - 1, and the eighth row contains numbers that differ by 4 from multiples of 25 = 6*4 + 1.

The first and second rows are A047209 and A047336.

The diagonal is A062717, the numbers x for which 6*x + 1 is a perfect square.

a(m,n) = 6*floor((m+1)/2)*floor((n+1)/2) + ((-1)^n)*floor((m+1)/2) + ((-1)^m)*floor((n+1)/2);
matrix(7, 7, n, k, a(n, k)) \\ Michel Marcus, Jan 25 2019

a(m,n) = 6*floor((m+1)/2)*floor((n+1)/2) + ((-1)^n)*floor((m+1)/2) + ((-1)^m)*floor((n+1)/2), m,n >= 1.

A074226 Numbers n such that Kronecker(3,n) = 1.

Original entry on oeis.org

1, 4, 10, 11, 13, 14, 16, 23, 25, 34, 35, 37, 38, 40, 44, 47, 49, 52, 56, 58, 59, 61, 62, 64, 71, 73, 82, 83, 85, 86, 92, 95, 97, 100, 106, 107, 109, 110, 119, 121, 130, 131, 133, 134, 136, 140, 143, 145, 148, 152, 154, 155, 157, 158, 160, 167, 169, 176, 178, 179

Offset: 1

Jon Perry, Sep 17 2002

nonn

Numbers n such that Kronecker(3, n) = mu(gcd(n, 3)).

Indranil Ghosh, Table of n, a(n) for n = 1..1666

Cf. A074227, A047209, A074229.

Select[Range@ 180, KroneckerSymbol[3, #]== 1 &]  (* Indranil Ghosh, Mar 16 2017 *)

for (x=1,200, for (y=1,200,if (kronecker(x,y)==moebius(gcd(x,y)),write("km.txt",x,";",y," : ",kronecker(x,y)))))

is(n)=kronecker(3,n)>0 \\ Charles R Greathouse IV, Apr 06 2012

a(n) seems to be asymptotic to c*n with c=3. - Benoit Cloitre, Sep 18 2002

Definition simplified by Charles R Greathouse IV, Apr 06 2012

Offset changed by Sean A. Irvine, Jan 14 2025

A319168 Frobenius pseudoprimes == 1,4 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

Original entry on oeis.org

4181, 6721, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 90061, 96049, 97921, 118441, 146611, 163081, 186961, 197209, 219781, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 330929, 399001, 433621, 438751, 489601, 512461, 520801

Offset: 1

Jianing Song, Sep 12 2018

nonn

Complement of A212423 with respect to A212424.
Intersection of A212424 and A047209.
Composite k == 1,4 (mod 5) such that Fibonacci(k) == 1 (mod k) and that k divides Fibonacci(k-1).

			4181 = 37*113 is composite, while Fibonacci(4180) == 0 (mod 4181), Fibonacci(4181) == 1 (mod 4181), so 4181 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (from Dana Jacobsen's site)
Jon Grantham, Frobenius pseudoprimes, Mathematics of Computation 70 (234): 873-891, 2001. doi: 10.1090/S0025-5718-00-01197-2.
Dana Jacobsen, Pseudoprime Statistics, Tables, and Data.
A. Rotkiewicz, Lucas and Frobenius Pseudoprimes, Annales Mathematicae Silesiane, 17 (2003): 17-39.
Eric Weisstein's World of Mathematics, Frobenius Pseudoprime.

Cf. A047209, A093372, A094394, A094401, A212423, A212424.

for(n=2,500000,if(!isprime(n) && (n%5==1||n%5==4) && fibonacci(n-kronecker(5,n))%n==0 && (fibonacci(n)-kronecker(5,n))%n==0, print1(n, ", ")))

A348600 Triangle read by rows: T(n,k) is the number of (unlabeled) connected graphs with n nodes and metric dimension k, 0 <= k < n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 13, 6, 1, 0, 1, 62, 39, 9, 1, 0, 1, 275, 488, 77, 11, 1, 0, 1, 1710, 8116, 1145, 130, 14, 1, 0, 1, 12061, 216432, 29958, 2415, 196, 16, 1, 0, 1, 93706, 9512947, 2026922, 78265, 4434, 276, 19, 1

Offset: 1

Pontus von Brömssen, Jan 26 2022

			Triangle begins:
  n\k| 0  1     2       3       4     5    6   7  8  9
  ---+------------------------------------------------
   1 | 1
   2 | 0  1
   3 | 0  1     1
   4 | 0  1     4       1
   5 | 0  1    13       6       1
   6 | 0  1    62      39       9     1
   7 | 0  1   275     488      77    11    1
   8 | 0  1  1710    8116    1145   130   14   1
   9 | 0  1 12061  216432   29958  2415  196  16  1
  10 | 0  1 93706 9512947 2026922 78265 4434 276 19  1

Gary Chartrand, Linda Eroh, Mark A. Johnson, and Ortrud R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Applied Mathematics 105 (2000), 99-113.
Richard C. Tillquist, Rafael M. Frongillo, and Manuel E. Lladser, Getting the lay of the land in discrete space: a survey of metric dimension and its applications, arXiv:2104.07201 [math.CO], 2021.
Wikipedia, Metric dimension

Cf. A047209, A303735.

Row sums: A001349.

T(n,1) = 1 for n >= 2, because the only graphs with metric dimension 1 are the paths of positive lengths (Chartrand et al. 2000).
T(n,n-2) = A047209(n-2) = floor(5*n/2-6) for n >= 3 (follows from the complete description of graphs with n nodes and metric dimension n-2 by Chartrand et al. 2000).
T(n,n-1) = 1 for n >= 1 , because the only graph with n nodes and metric dimension n-1 is the complete graph (Chartrand et al. 2000).

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A323674 Square array, read by antidiagonals, of the positive integers 6cd +-c +-d = (6c +- 1)d +- c. Alternate rows (or columns) are numbers that differ by c from multiples of 6c - 1 or 6c + 1.

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A074226 Numbers n such that Kronecker(3,n) = 1.

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A319168 Frobenius pseudoprimes == 1,4 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

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A348600 Triangle read by rows: T(n,k) is the number of (unlabeled) connected graphs with n nodes and metric dimension k, 0 <= k < n.

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