may has authored 29383 sequences. Here are the ten most recent ones:
A383998
Number of distinct truncated Graham sequences (as in Graham's Tree Reconstruction Conjecture) of length 4 on trees of order n.
Original entry on oeis.org
1, 1, 1, 2, 3, 6, 11, 20, 37, 68, 114, 188, 300, 462, 702, 1041
Offset: 1
For n=8, there are 23 trees but only 20 distinct truncated Graham sequences of length 4.
There are two pairs of trees on 8 vertices which have the same length-4 sequence [|G|, |L(G)|, |L(L(G))|, |L(L(L(G)))|], namely the sequence [8,7,7,9] which comes from both the (unlabeled versions of) {{1, 2}, {2, 3}, {3, 7}, {4, 5}, {5, 6}, {6, 7}, {7, 8}} and {{1, 2}, {2, 3}, {3, 8}, {4, 7}, {5, 6}, {6, 7}, {7, 8}}.
But for sequences of length 5 there are different sequences, namely
[8, 7, 7, 9, 18] and [8, 7, 7, 9, 17]: the sequence [8,7,9,17] comes from both {{1, 3}, {2, 3}, {3, 7}, {4, 6}, {5, 6}, {6, 7}, {7, 8}} and {{1, 2}, {2, 3}, {3, 8}, {4, 7}, {5, 7}, {6, 7}, {7, 8}}
So Graham's conjecture is confirmed for trees with 8 vertices, but requires using sequences of length up to 5.
A384342
Largest minimum height of the irreducible factors of a degree-n polynomial of height 1.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2
Offset: 1
For n < 12, every height 1 degree n polynomial has a height 1 irreducible factor, so a(n) = 1.
For n = 12, x^12-x^11-x^9-x^8+x^6-x^4+x^3+x+1 = (x^6-2x^5+x^4-x^2+x-1)(x^6+x^5+x^4-x^2-2x-1) is the product of two irreducible polynomials of height 2, so a(12) >= 2; and every degree 12 height 1 polynomial has an irreducible factor of height at most 2, so a(12) = 2.
Cf.
A363959 gives max height of max-height irreducible factor, whereas this sequence gives max height of min-height irreducible factor.
-
from msmath.poly import polynomial as poly
def height(p) :
"""find the height, i.e. max abs coeff, of poly p"""
return max(map(abs, p));
def height1(n) :
"""generate all height 1 polys of degree n"""
for a in range(3**n) :
p = [1];
for i in range(n) :
a, r = divmod(a, 3);
p.append(r-1);
yield poly(*p);
def a(n) :
"""Return max min height of the irreducible factors of a degree n height 1 poly"""
highest = 0;
for p in height1(n) :
f = p.factor();
h = min(map(height, f));
if highest < h:
highest = h;
return highest;
A383823
a(n) is the number of secondary GL(4) invariants contructed from n+3 distinct four component vectors.
Original entry on oeis.org
1, 5, 105, 4116, 232848, 16818516, 1447482465, 142174944340, 15484613937936, 1832516612010448, 232187445047217296, 31148053701600494400
Offset: 1
- Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
A382350
Number of maximal antichains in the Bruhat order on B_n.
Original entry on oeis.org
2, 5, 215, 24828398365
Offset: 1
For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The maximal antichains are {1} and {s1}.
For n=2 the line (Hasse) diagram is below.
s2*s1*s2*s1
/ \
s2*s1*s2 s1*s2*s1
| X |
s2*s1 s1*s2
| X |
s2 s1
\ /
1
The set of maximal antichains is {{1}, {s2, s1}, {s2*s1, s1*s2}, {s2*s1*s2, s1*s2*s1}, {s2*s1*s2*s1}}.
- A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
Cf.
A382346 (antichains),
A005900 (the number of join-irreducible elements),
A378072 (the size of Dedekind-MacNeille completion)
A382751
Numbers k for which the 3-adic valuation A007949(k) == 0 (mod 3).
Original entry on oeis.org
1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 54, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95
Offset: 1
7 is a term since its 3-adic valuation is A007949(7) = 0 which is == 0 (mod 3).
-
Select[Range[100], Divisible[IntegerExponent[#, 3], 3] &] (* Amiram Eldar, May 13 2025 *)
-
isok(k) = (valuation(k, 3) % 3) == 0; \\ Michel Marcus, Jun 03 2025
A384002
Let S(n,j,k), j = 1..n, k = 1..A024718(n), where row 1 = {(0),(1)}, and row n = union of n-tuples whose sum m < n, and the n-tuples formed by appending m to the (n-1)-tuples in row n-1. Then T(n,j) = j-th tuple in row n of S read as a base n+1 number expressed in decimal.
Original entry on oeis.org
0, 1, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 6, 8, 9, 16, 17, 18, 20, 21, 32, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15, 16, 25, 26, 27, 28, 30, 31, 32, 35, 36, 50, 51, 52, 55, 56, 75, 125, 126, 127, 128, 130, 131, 132, 135, 136, 150, 151, 152, 155, 156, 175, 250, 251, 252, 255, 275, 375
Offset: 1
Table begins:
1: 0, 1;
2: 0, 1, 2, 3, 4;
3: 0, 1, 2, 3, 4, 5, 6, 8, 9, 16, 17, 18, 20, 21, 32;
4: 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15, 16, 25, 26, 27, 28, 30, 31, 32,
35, 36, 50, 51, 52, 55, 56, 75, 125, 126, 127, 128, 130, 131, 132, 135,
136, 150, 151, 152, 155, 156, 175, 250, 251, 252, 255, 275, 375;
etc.
Row 2 of S is {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)}. Reading the tuples in row 2 as a base 3 number, we have row 2 of this sequence.
-
nn = 8; w[0] = {{0}};
Do[If[n == 1, Set[w[1], {{0}, {1}}],
Set[w[n], Union@ Join[Select[Tuples[Range[0, n - 1], n], Total[#] < n &],
Map[Append[#, n - Total[#]] &, w[n - 1] ] ] ] ], {n, nn}];
Table[Map[FromDigits[#, n + 1] &, w[n]], {n, 0, nn}]
A384001
Irregular triangle T(n,j,k), j = 1..A024718(n), k = 1..n, where row 1 = {(0), (1)}, and row n = union of n-tuples whose sum s < n, and the n-tuples formed by appending s to the (n-1)-tuples in row n-1.
Original entry on oeis.org
0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 2, 0, 0, 2, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2
Offset: 1
Table begins:
1: (0), (1);
2: (0, 0), (0, 1), (0, 2), (1, 0), (1, 1);
3: (0,0,0), (0,0,1), (0,0,2), (0,0,3), (0,1,0),
(0,1,1), (0,1,2), (0,2,0), (0,2,1), (1,0,0),
(1,0,1), (1,0,2), (1,1,0), (1,1,1), (2,0,0)
etc.
Row 2 arranged as a rank 2 table, concatenating T(2,j,k), k = 1..2:
00 10 20
01 11
.
Row 3 arranged as a rank 3 table, concatenating T(3,j,k), k = 1..3:
000 001 002 003 100 101 102 200
010 011 012 110 111
020 021
-
nn = 4; w[0] = {{0}};
Do[If[n == 1, Set[w[1], {{0}, {1}}],
Set[w[n], Union@ Join[Select[Tuples[Range[0, n - 1], n], Total[#] < n &],
Map[Append[#, n - Total[#]] &, w[n - 1] ] ] ] ], {n, nn}];
Flatten@ Array[w, nn]
A383594
a(0) = 0 and thereafter a(n) = 2 if a(n-1) is an odd prime, otherwise a(n) = a(n-1) + k where k = n - P(n) and P(n) is the number of odd primes among terms a(0),...,a(n-1).
Original entry on oeis.org
0, 1, 3, 2, 5, 2, 6, 11, 2, 8, 15, 23, 2, 11, 2, 12, 23, 2, 14, 27, 41, 2, 17, 2, 18, 35, 53, 2, 21, 41, 2, 23, 2, 24, 47, 2, 26, 51, 77, 104, 132, 161, 191, 2, 33, 65, 98, 132, 167, 2, 38, 75, 113, 2, 41, 2, 42, 83, 2, 44, 87, 131, 2, 47, 2, 48, 95, 143, 192, 242
Offset: 0
The sequence, and the k increments applied, begin
a(n) = 0, 1, 3, 2, 5, 2, 6, 11, 2, 8, 15, 23, ...
k = 1 2 3 4 5 6 7 8
-
seq := proc(n)
local a, i, k;
a := [0];
k := 1;
for i from 1 to n-1 do
if isprime(a[-1]) and a[-1] <> 2 then
a := [op(a), 2];
else
a := [op(a), a[-1] + k];
k := k + 1;
end if;
end do;
return a;
end proc:
-
sequence[n_] := Module[{a = {0}, k = 1},
While[Length[a] < n,
If[PrimeQ[Last[a]] && Last[a] != 2,
AppendTo[a, 2],
AppendTo[a, Last[a] + k];
k++
];
];
a
]
A384033
a(n) is the number of solutions to n = sopfr(k*sopfr(n)) where sopfr(m) is sum of prime factors of m counted with multiplicity.
Original entry on oeis.org
0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 4, 5, 1, 7, 1, 10, 10, 4, 1, 19, 19, 19, 30, 17, 1, 40, 1, 52, 46, 12, 77, 87, 1, 77, 111, 87, 1, 175, 1, 197, 157, 197, 1, 219, 302, 413, 372, 175, 1, 413, 614, 413, 456, 67, 1, 1083, 1, 677, 819, 1552, 1552, 1695, 1
Offset: 1
A star graph can be defined differently; here, the star graph S_k has a total of k vertices and k-1 leaves.
For all primes n=p as leaf label, the star graph S_2 with 1 central vertex labeled p and 1 leaf labeled p is a solution and the only solution; thus, a(p)=1.
p - p
n=4 as leaf label, works as well with S_2 (2+2=2*2); thus, a(4)=1.
4 - 4
For n=6 as leaf label, there is no respective star graph (as the sum of prime factors is 2+3=5).
n=8 as leaf label can be done as S_4 with the center labeled 18 and the leaves labeled 8 (the prime factor sum for 18=2*3^2 is 2+3+3=8; the prime factor sum for 8=2^3 is 6, adding over all 3 leaves, results in 18); thus, a(8)=1.
8
|
8 - 18 - 8
n=12 is the first labeling with 2 solutions: it can be done with central vertex labeled 35 (S_6) and 42 (S_7); thus, a(12)=2.
The S_6 solution looks like
12 12
\ /
12 - 35 - 12
|
12
as 12=2*2*3 -> 7, 5 leaves sum up to 35 = 5*7 -> 12.
-
sopfr[n_] := sopfr[n] = Plus @@ Times @@@ FactorInteger@ n; sopfr[1] = 0; t = Series[1/Product[1 - x^Prime[i], {i, 100}], {x, 0, 100}]; a[n_] := Block[{s = sopfr@sopfr@n}, If[s <= n, Coefficient[t, x, n - s], 0]]; Array[a, 67] (* - Robert G. Wilson v, Jul 16 2025 *)
A383858
Irregular triangle read by rows: T(n,k) (n >= 4, 4 <= k <= A384502(n)) is the smallest n-digit number m with k distinct prime factors, such that these factors can be divided into two subsets of at least two elements each, both summing to the same value. If no such number exists, T(n,k) = -1.
Original entry on oeis.org
2145, 2310, 10725, 10374, 101065, 100050, 255255, 510510, 1005993, 1000350, 1036035, 1009470, 10006081, 10000130, 10012065, 10004610, 100010225, 100001300, 100001195, 100009910, 111546435, 223092870, 1000083889, 1000008758, 1000001751, 1000005270, 1002569295, 1001110110
Offset: 4
T(4,4) = 2145 = 3*5*11*13 is the smallest four-digit number with four distinct prime factors (3, 5, 11, and 13), where the prime factors can be partitioned into two subsets of at least two elements each, both summing to the same value: 3+13 = 5+11.
T(5,4) = 2310 = 2*3*5*7*11 is the smallest five-digit number with four distinct prime factors (2, 3, 5, 7 and 11), where the prime factors can be partitioned into two subsets of at least two elements each, both summing to the same value: 3+11 = 2+5+7.
The lower triangle begins at T(4,4):
[ 2145, 2310];
[ 10725, 10374];
[ 101065, 100050, 255255, 510510];
[ 1005993, 1000350, 1036035, 1009470];
[ 10006081, 10000130, 10012065, 10004610];
[100010225, 100001300, 100001195, 100009910, 111546435, 223092870]; ...
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