A322804
Numbers that can be written as a product of one or more consecutive primorial numbers.
Original entry on oeis.org
1, 2, 6, 12, 30, 180, 210, 360, 2310, 6300, 30030, 37800, 75600, 485100, 510510, 9699690, 14553000, 69369300, 87318000, 174636000, 223092870, 6469693230, 14567553000, 15330615300, 200560490130, 437026590000, 2622159540000, 4951788741900, 5244319080000
Offset: 1
A329894
Terms of A025487 from which the distance to the next larger prime is a composite number.
Original entry on oeis.org
512, 16384, 373248, 393216, 524288, 1119744, 4194304, 4718592, 5971968, 8388608, 10077696, 10616832, 17915904, 21233664, 31104000, 33554432, 35831808, 42467328, 47775744, 56623104, 67108864, 150994944, 159252480, 286654464, 322486272, 362797056, 679477248, 859963392, 1528823808, 2176782336, 2890137600, 4294967296, 5804752896, 8748000000
Offset: 1
As A151800(512) = 521, with 521 - 512 = 9 (a composite number), 512 is included in this sequence.
-
isc(n) = ((n > 1)&&!isprime(n));
for(n=1,2000,if(isc(nextprime(1+A025487(n))-A025487(n)),print1(A025487(n),", ")));
A334175
Numbers that can be written as a product of two or more consecutive primorial numbers.
Original entry on oeis.org
2, 12, 180, 360, 6300, 37800, 75600, 485100, 14553000, 69369300, 87318000, 174636000, 14567553000, 15330615300, 437026590000, 2622159540000, 4951788741900, 5244319080000, 35413721343000, 2163931680210300, 7436881482030000, 148702215919257000, 223106444460900000
Offset: 1
2 = prime(0)# * prime(1)#;
12 = prime(1)# * prime(2)#;
180 = prime(2)# * prime(3)#;
360 = prime(1)# * prime(2)# * prime(3)#;
6300 = prime(3)# * prime(4)#,
where prime(k)# is the product of the first k primes.
A350424
Numbers for which the number of their semiprime divisors sets a new record.
Original entry on oeis.org
4, 12, 30, 60, 180, 210, 420, 1260, 2310, 4620, 13860, 30030, 60060, 180180, 510510, 1021020, 3063060, 9699690, 19399380, 58198140, 223092870, 446185740, 1338557220, 6469693230, 12939386460, 38818159380, 194090796900, 200560490130, 401120980260, 1203362940780, 6016814703900
Offset: 1
A350425 gives the corresponding number of semiprime divisors.
A363458
Numbers k such that k and k+1 are both in A363457.
Original entry on oeis.org
1, 54, 242883, 246962, 261643, 266001, 353893, 380287, 425818, 457055, 542950, 581942, 595440, 831264, 917311, 980235, 1256341, 1719654, 6239931, 8237549, 8378312, 10995744, 11650985, 15123420, 15194370, 15442721, 19628056, 20034738, 20308106, 26218271, 36099782
Offset: 1
54 is a term since A025487(54) = 2310 and A025487(55) = 2520 are both products of distinct primorial numbers: 2310 = 2 * 3 * 5 * 7 * 11 and 2520 = 2 * (2 * 3) * (2 * 3 * 5 * 7).
A383733
Number of proper 3-colorings of the generalized chorded cycle graph C_n^{(3)}.
Original entry on oeis.org
42, 0, 0, 18, 186, 66, 0, 234, 930, 750, 0, 2244, 4578, 6498, 120
Offset: 6
For n=6, consider the graph C_6^(3), constructed as follows:
- Start with a cycle graph (hexagon) having vertices labeled {0,1,2,3,4,5}.
- Add chords connecting vertex i with vertex i+3 mod 6, forming edges (0,3), (1,4), (2,5).
- Since n is even, include diametric edges connecting opposite vertices: edges (0,3), (1,4), (2,5). (Note these diametric edges coincide with chords for n=6.)
The resulting graph is symmetric and moderately dense. Enumerating explicitly all possible vertex-coloring assignments with exactly three colors, we find precisely 42 distinct valid 3-colorings (each satisfying the condition that no two adjacent vertices share the same color).
Thus, a(6)=42.
- N. L. Biggs, Algebraic Graph Theory. Cambridge University Press, 2nd ed., 1993.
- D. B. West, Introduction to Graph Theory. Prentice Hall, 2nd ed., 2001.
- R. J. Wilson, Graph Theory. Longman, 5th impression, 1996.
Cf.
A000670 (number of preferential arrangements),
A001047 (chromatic polynomial of cycles at x=3),
A003049 (chromatic polynomial of complete graphs),
A129912 (number of 3-colorings of certain circulant graphs).
Related to chromatic polynomial evaluations and modular coloring patterns not captured by standard families. May also be compared to sequences involving nonzero chromatic roots and Beraha numbers.
-
with(GraphTheory):
Cn3_graph := proc(n)
local G, i;
G := CycleGraph(n);
for i from 0 to n-1 do
AddEdge(G, {i, (i+3) mod n});
end do;
if modp(n, 2) = 0 then
for i from 0 to n/2 - 1 do
AddEdge(G, {i, (i + n/2) mod n});
end do;
end if;
return G;
end proc:
a := proc(n) local G;
G := Cn3_graph(n);
return ChromaticPolynomial(G, 3);
end proc:
# Compute initial terms from n=6 to n=20:
seq(a(n), n=6..20);
-
Cn3Graph[n_] := Module[{g, edges, i},
edges = Table[{i, Mod[i + 1, n]}, {i, 0, n - 1}]; (* Cycle edges *)
edges = Join[edges, Table[{i, Mod[i + 3, n]}, {i, 0, n - 1}]]; (* Chord edges *)
If[EvenQ[n],
edges = Join[edges, Table[{i, Mod[i + n/2, n]}, {i, 0, n/2 - 1}]]
];
Graph[edges, VertexLabels -> "Name"]
];
a[n_] := Length@Select[
Tuples[{1, 2, 3}, n],
And @@ (#[[#[[1]] + 1]] != #[[#[[2]] + 1]] & /@
EdgeList[Cn3Graph[n]] /. {x_, y_} :> {x, y})
] &;
(* Generate terms for n from 6 to 20 *)
Table[a[n], {n, 6, 20}]
-
# Illustrative brute-force check for small n using networkx
import networkx as nx
from itertools import product
def Cn_k_graph(n, k):
G = nx.cycle_graph(n)
for i in range(n):
G.add_edge(i, (i+k)%n)
if n % 2 == 0:
for i in range(n//2):
G.add_edge(i, i+n//2)
return G
def count_colorings(G, colors=3):
nodes = list(G.nodes())
count = 0
for coloring in product(range(colors), repeat=len(nodes)):
if all(coloring[u] != coloring[v] for u,v in G.edges()):
count += 1
return count
# Example usage:
for n in range(6, 21):
G = Cn_k_graph(n, 3)
print(f'n={n}, colorings={count_colorings(G)}')
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