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Displaying 1-10 of 357 results found. page 1 2 3 4 5 6 7 8 9 10 ... 36
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A371970 Exponents k such that the binary expansion of 3^k has an even number of ones. +0
0
1, 2, 3, 5, 6, 8, 9, 12, 14, 17, 18, 21, 23, 24, 25, 26, 27, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 47, 51, 52, 55, 57, 58, 59, 60, 61, 64, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 96, 99, 102, 104, 105, 106, 109, 112, 116, 127, 131, 132, 133, 134, 135, 136 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Hugo Pfoertner, Table of n, a(n) for n = 1..10000
MAPLE
q:= n-> is(add(i, i=Bits[Split](3^n))::even):
select(q, [$0..150])[]; # Alois P. Heinz, Apr 24 2024
MATHEMATICA
Select[Range[136], EvenQ@ DigitCount[3^#, 2, 1] &] (* Michael De Vlieger, Apr 24 2024 *)
PROG
(PARI) is_a371970(k) = hammingweight(3^k)%2 == 0
CROSSREFS
Complement of A223024.
Cf. A000120, A000244, A001969, A011754, A078839.
KEYWORD
nonn,base,easy,new
AUTHOR
Hugo Pfoertner, Apr 24 2024
STATUS
approved
A372237 a(0) = 4; to obtain a(k), write out the base-(2^k) expansion of a(k-1), bump to base 2^(k+1), then subtract 1. +0
0
4, 15, 26, 49, 96, 191, 318, 573, 1084, 2107, 4154, 8249, 16440, 32823, 65590, 131125, 262196, 524339, 1048626, 2097201, 4194352, 8388655, 16777262, 33554477, 67108908, 134217771, 268435498, 536870953, 1073741864, 2147483687, 4294967334, 8589934629, 17179869220 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Applying to the proof of the usual Goodstein's theorem to the ordinal number omega^omega shows that: for no matter what initial value and no matter what increasing sequence of bases b(0), b(1), ... with b(0) >= 2, the (weak) Goodstein sequence eventually terminates with 0. Here b(k) = 2^(k+1).
Sequence terminates at a(2^(2^70+70) + 2^70 + 68) = 0.
LINKS
Table of n, a(n) for n=0..32.
Googology Wiki, Goodstein sequence.
Wikipedia, Goodstein's Theorem
FORMULA
a(k) = 2^(k+2) + 68 - k for 5 <= k <= 68. The base-(2^(k+1)) expansion of a(k) consists of two digits 2 and 68 - k.
a(k) = 2^(k+1) + 2^70 + 68 - k for 69 <= 2^70 + 68. The base-(2^(k+1)) expansion of a(k) consists of two digits 1 and 2^70 + 68 - k.
a(k) = 2^(2^70+70) + 2^70 + 68 - k for 2^70 + 69 <= k <= 2^(2^70+70) + 2^70 + 68. The base-(2^(k+1)) expansion of a(k) consists of a single digit 2^(2^70+70) + 2^70 + 68 - k.
EXAMPLE
a(0) = 100_2 = 4;
a(1) = 100_4 - 1 = 15 = 33_4;
a(2) = 33_8 - 1 = 26 = 32_8;
a(3) = 32_16 - 1 = 49 = 31_16;
a(4) = 31_32 - 1 = 96 = 30_32;
a(5) = 30_64 - 1 = 191 = (2,63)_64.
PROG
(PARI) A372237_first_N_terms(N) = my(v=vector(N+1)); v[1] = 4; for(i=1, N, v[i+1] = fromdigits(digits(v[i], 2^i), 2^(i+1))-1); v
CROSSREFS
Cf. A056193, A266202.
KEYWORD
nonn,easy,fini,new
AUTHOR
Jianing Song, Apr 23 2024
STATUS
approved
A372261 Number T(n,k,j) of acyclic orientations of the complete tripartite graph K_{n,k,j}; triangle of triangles T(n,k,j), n>=0, k=0..n, j=0..k, read by rows. +0
0
1, 1, 2, 6, 1, 4, 18, 14, 78, 426, 1, 8, 54, 46, 330, 2286, 230, 1902, 15402, 122190, 1, 16, 162, 146, 1374, 12090, 1066, 10554, 101502, 951546, 6902, 76110, 822954, 8724078, 90768378, 1, 32, 486, 454, 5658, 63198, 4718, 57054, 657210, 7290942, 41506, 525642, 6495534, 78463434, 928340190 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.
LINKS
Alois P. Heinz, Rows n = 0..40, flattened
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
Wikipedia, Acyclic orientation
Wikipedia, Multipartite graph
EXAMPLE
Triangle of triangles T(n,k,j) begins:
1;
;
1;
2, 6;
;
1;
4, 18;
14, 78, 426;
;
1;
8, 54;
46, 330, 2286;
230, 1902, 15402, 122190;
;
...
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(
expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
end:
T:= proc() option remember; local q, l, b; q, l, b:= -1, [args],
proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*
(q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))
end; abs(b(0, nops(l)))
end:
seq(seq(seq(T(n, k, j), j=0..k), k=0..n), n=0..5);
CROSSREFS
T(n,k,0) for k=0..9 give: A000012, A000079, A027649, A027650, A027651, A283811, A283812, A283813, A284032, A284033.
T(n,n,n) gives A370961.
T(n,n,0) gives A048163(n+1).
T(n+1,n,0) gives A188634(n+1).
T(n,1,1) gives A008776.
T(n,2,2) gives A370960.
Cf. A099594, A267383, A372254.
KEYWORD
nonn,look,tabf,new
AUTHOR
Alois P. Heinz, Apr 24 2024
STATUS
approved
A372254 Number A(n,k) of acyclic orientations of the complete tripartite graph K_{n,n,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals. +0
0
1, 1, 2, 1, 6, 14, 1, 18, 78, 230, 1, 54, 426, 1902, 6902, 1, 162, 2286, 15402, 76110, 329462, 1, 486, 12090, 122190, 822954, 4553166, 22934774, 1, 1458, 63198, 951546, 8724078, 61796298, 381523758, 2193664790, 1, 4374, 327306, 7290942, 90768378, 823457454, 6241779786, 42700751022, 276054834902 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
Wikipedia, Acyclic orientation
Wikipedia, Multipartite graph
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
2, 6, 18, 54, 162, 486, ...
14, 78, 426, 2286, 12090, 63198, ...
230, 1902, 15402, 122190, 951546, 7290942, ...
6902, 76110, 822954, 8724078, 90768378, 928340190, ...
329462, 4553166, 61796298, 823457454, 10779805722, 138779942046, ...
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(
expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
end:
A:= proc(n, k) option remember; local q, l, b; q, l, b:= -1, [n$2, k],
proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*
(q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))
end; abs(b(0, 3))
end:
seq(seq(A(n, d-n), n=0..d), d=0..9);
CROSSREFS
Rows n=0-2 give: A000012, A008776, A370960.
Column k=0 gives A048163(n+1).
Main diagonal gives A370961.
Cf. A267383, A372261.
KEYWORD
nonn,tabl,new
AUTHOR
Alois P. Heinz, Apr 24 2024
STATUS
approved
A371097 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = A371092(A371095(n, k)), n,k >= 1. +0
0
1, 1, 2, 1, 2, 3, 1, 3, 1, 4, 1, 3, 1, 5, 5, 1, 1, 1, 2, 4, 6, 1, 1, 1, 3, 5, 8, 7, 1, 1, 1, 3, 2, 12, 2, 8, 1, 1, 1, 1, 3, 18, 2, 11, 9, 1, 1, 1, 1, 3, 27, 3, 9, 7, 10, 1, 1, 1, 1, 1, 21, 3, 7, 2, 14, 11, 1, 1, 1, 1, 1, 16, 1, 2, 2, 21, 4, 12, 1, 1, 1, 1, 1, 23, 1, 2, 3, 8, 6, 17, 13, 1, 1, 1, 1, 1, 18, 1, 3, 3, 12, 9, 4, 10, 14 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
A(n, k) gives the column index of A371095(n, k) [or equally, of A371096(n, k)] in array A257852.
LINKS
Table of n, a(n) for n=1..105.
Index entries for sequences related to 3x+1 (or Collatz) problem
FORMULA
A(n, k) = A371092(A371095(n, k)) = A371092(A371096(n, k)).
EXAMPLE
Array begins:
n\k| 1 2 3 ...
---+--------------------------------------------------------------
1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
2 | 1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20, 1, 23, 13,
3 | 1, 3, 1, 2, 5, 12, 2, 9, 2, 21, 6, 4, 14, 30, 1, 18, 10,
4 | 1, 3, 1, 3, 2, 18, 3, 7, 2, 8, 9, 5, 21, 45, 1, 26, 14,
5 | 1, 1, 1, 3, 3, 27, 3, 2, 3, 12, 1, 2, 8, 17, 1, 39, 21,
6 | 1, 1, 1, 1, 3, 21, 1, 2, 3, 18, 1, 3, 12, 4, 1, 30, 8,
7 | 1, 1, 1, 1, 1, 16, 1, 3, 1, 27, 1, 3, 18, 5, 1, 44, 12,
8 | 1, 1, 1, 1, 1, 23, 1, 3, 1, 21, 1, 1, 27, 2, 1, 66, 18,
9 | 1, 1, 1, 1, 1, 18, 1, 1, 1, 16, 1, 1, 21, 3, 1, 99, 27,
10 | 1, 1, 1, 1, 1, 26, 1, 1, 1, 23, 1, 1, 16, 3, 1, 75, 21,
11 | 1, 1, 1, 1, 1, 39, 1, 1, 1, 18, 1, 1, 23, 1, 1, 28, 16,
12 | 1, 1, 1, 1, 1, 30, 1, 1, 1, 26, 1, 1, 18, 1, 1, 42, 23,
13 | 1, 1, 1, 1, 1, 44, 1, 1, 1, 39, 1, 1, 26, 1, 1, 63, 18,
14 | 1, 1, 1, 1, 1, 66, 1, 1, 1, 30, 1, 1, 39, 1, 1, 48, 26,
15 | 1, 1, 1, 1, 1, 99, 1, 1, 1, 44, 1, 1, 30, 1, 1, 71, 39,
16 | 1, 1, 1, 1, 1, 75, 1, 1, 1, 66, 1, 1, 44, 1, 1, 54, 30,
17 | 1, 1, 1, 1, 1, 28, 1, 1, 1, 99, 1, 1, 66, 1, 1, 80, 44,
18 | 1, 1, 1, 1, 1, 42, 1, 1, 1, 75, 1, 1, 99, 1, 1, 120, 66,
19 | 1, 1, 1, 1, 1, 63, 1, 1, 1, 28, 1, 1, 75, 1, 1, 180, 99,
20 | 1, 1, 1, 1, 1, 48, 1, 1, 1, 42, 1, 1, 28, 1, 1, 270, 75,
21 | 1, 1, 1, 1, 1, 71, 1, 1, 1, 63, 1, 1, 42, 1, 1, 405, 28,
PROG
(PARI)
up_to = 105;
A000265(n) = (n>>valuation(n, 2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A371095sq(n, k) = if(1==n, 8*k-7, R(A371095sq(n-1, k)));
A371097sq(n, k) = A371092(A371095sq(n, k));
A371097list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A371097sq((a-(col-1)), col))); (v); };
v371097 = A371097list(up_to);
A371097(n) = v371097[n];
CROSSREFS
Cf. A000027 (row 1), A257852, A371092, A371095, A371096.
Cf. also arrays A371101, A371103.
KEYWORD
nonn,tabl,easy,new
AUTHOR
Antti Karttunen, Apr 21 2024
STATUS
approved
A371095 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 8*k-7, and A(n+1, k) = R(A(n, k)), n,k >= 1, where Reduced Collatz function R(n) gives the odd part of 3n+1. +0
0
1, 1, 9, 1, 7, 17, 1, 11, 13, 25, 1, 17, 5, 19, 33, 1, 13, 1, 29, 25, 41, 1, 5, 1, 11, 19, 31, 49, 1, 1, 1, 17, 29, 47, 37, 57, 1, 1, 1, 13, 11, 71, 7, 43, 65, 1, 1, 1, 5, 17, 107, 11, 65, 49, 73, 1, 1, 1, 1, 13, 161, 17, 49, 37, 55, 81, 1, 1, 1, 1, 5, 121, 13, 37, 7, 83, 61, 89, 1, 1, 1, 1, 1, 91, 5, 7, 11, 125, 23, 67, 97 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..91.
Index entries for sequences related to 3x+1 (or Collatz) problem
EXAMPLE
Array begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---+------------------------------------------------------------------------
1 | 1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 121,
2 | 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91,
3 | 1, 11, 5, 29, 19, 47, 7, 65, 37, 83, 23, 101, 55, 119, 1, 137,
4 | 1, 17, 1, 11, 29, 71, 11, 49, 7, 125, 35, 19, 83, 179, 1, 103,
5 | 1, 13, 1, 17, 11, 107, 17, 37, 11, 47, 53, 29, 125, 269, 1, 155,
6 | 1, 5, 1, 13, 17, 161, 13, 7, 17, 71, 5, 11, 47, 101, 1, 233,
7 | 1, 1, 1, 5, 13, 121, 5, 11, 13, 107, 1, 17, 71, 19, 1, 175,
8 | 1, 1, 1, 1, 5, 91, 1, 17, 5, 161, 1, 13, 107, 29, 1, 263,
9 | 1, 1, 1, 1, 1, 137, 1, 13, 1, 121, 1, 5, 161, 11, 1, 395,
10 | 1, 1, 1, 1, 1, 103, 1, 5, 1, 91, 1, 1, 121, 17, 1, 593,
11 | 1, 1, 1, 1, 1, 155, 1, 1, 1, 137, 1, 1, 91, 13, 1, 445,
12 | 1, 1, 1, 1, 1, 233, 1, 1, 1, 103, 1, 1, 137, 5, 1, 167,
13 | 1, 1, 1, 1, 1, 175, 1, 1, 1, 155, 1, 1, 103, 1, 1, 251,
14 | 1, 1, 1, 1, 1, 263, 1, 1, 1, 233, 1, 1, 155, 1, 1, 377,
15 | 1, 1, 1, 1, 1, 395, 1, 1, 1, 175, 1, 1, 233, 1, 1, 283,
16 | 1, 1, 1, 1, 1, 593, 1, 1, 1, 263, 1, 1, 175, 1, 1, 425,
PROG
(PARI)
up_to = 91;
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A371095sq(n, k) = if(1==n, 8*k-7, R(A371095sq(n-1, k)));
A371095list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A371095sq((a-(col-1)), col))); (v); };
v371095 = A371095list(up_to);
A371095(n) = v371095[n];
CROSSREFS
Cf. A017077 (row 1), A016921 (row 2), A075677.
Cf. also A371096, A371097.
KEYWORD
nonn,tabl,easy,new
AUTHOR
Antti Karttunen, Apr 24 2024
STATUS
approved
A372056 Smallest prime obtained by appending one or more 3's to n, or -1 if no such prime exists. +0
0
13, 23, -1, 43, 53, -1, 73, 83, -1, 103, 113, -1, 1333333333333333, 1433, -1, 163, 173, -1, 193, 20333, -1, 223, 233, -1, 2533333333, 263, -1, 283, 293, -1, 313, 323333, -1, 3433, 353, -1, 373, 383, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Next term is 40 followed by 483 3's and is too large to display here (see the b-file).
LINKS
Toshitaka Suzuki, Table of n, a(n) for n = 1..409
EXAMPLE
For n = 13, a(13) = 1333333333333333 is a prime (but 133,1333,13333 etc. are not primes).
CROSSREFS
See A112394 for another version.
Cf. A090584, A112386, A113076.
KEYWORD
base,sign,new
AUTHOR
Toshitaka Suzuki, Mar 30 2024
EXTENSIONS
Edited by N. J. A. Sloane, Apr 24 2024
STATUS
approved
A371602 Taxicab numbers that are sandwiched between squarefree numbers. +0
0
4104, 32832, 39312, 110808, 171288, 262656, 314496, 373464, 513000, 886464, 1016496, 1075032, 1195112, 1331064, 1370304, 1407672, 1609272, 1728216, 1734264, 1774656, 2101248, 2515968, 2864288, 2987712, 2991816, 3511872, 3512808, 3551112, 4104000, 4342914, 4467528, 4511808, 4607064 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All terms are even numbers.
LINKS
Table of n, a(n) for n=1..33.
Christian Boyer, Les nombres Taxicabs, in Dossier Pour La Science, pp. 26-28, Volume 59 (Jeux math') April/June 2008 Paris.
EXAMPLE
4104 = 2^3 * 3^3 * 19 (between 4103 = 11 * 373 and 4105 = 5 * 821).
32832 = 2^6 * 3^3 *19 (between 32831 and 32833, which are twin primes).
39312 = 2^4 * 3^3 * 7 * 13 (between 39311 = 19 * 2069 and 39313, which is prime).
MATHEMATICA
Select[Range[300000], And @@ SquareFreeQ /@ (# + {-1, 1}) && Length[PowersRepresentations[#, 2, 3]] > 1 &] (* Amiram Eldar, Mar 29 2024 *)
CROSSREFS
Intersection of A001235 and A067874.
A272892 is a subsequence.
Cf. A011541, A005117.
KEYWORD
nonn,new
AUTHOR
Massimo Kofler, Mar 29 2024
STATUS
approved
A371611 Number of 2*n-sided cycles with the property that one makes the same number of left and right turns while following its edges. +0
0
4, 36, 1240, 73240, 7171176, 1016813448, 198480110880, 50752206180576, 16460660622560680, 6595414427636900536, 3198428240666246044704, 1845848150787599809368856, 1250049326783769438348496480, 981653074459964543314138858320 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
Cycles that differ by rotation or reflection are counted separately. By "n-sided cycles" we mean the cycles that can be drawn by connecting n equally spaced points on a circle (possibly self-intersecting).
LINKS
Table of n, a(n) for n=2..15.
FORMULA
a(n) is always divisible by 2*n, because the considered polygons cannot have rotational symmetry.
CROSSREFS
Cf. A365094, A295264,
KEYWORD
nonn,new
AUTHOR
Ludovic Schwob, Mar 29 2024
STATUS
approved
A371598 a(n) = (Product_{i=1..n} Fibonacci(i)) mod Fibonacci(n + 1). +0
0
0, 1, 2, 1, 6, 6, 12, 2, 15, 16, 0, 49, 299, 220, 882, 252, 2176, 166, 495, 5720, 5251, 6065, 28224, 41650, 106947, 113288, 256737, 173841, 26840, 25379, 444150, 347278, 1834953, 8709610, 4046544, 2653673, 31127545, 47532000, 50717205, 147239197, 97769672, 37543458 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..42.
FORMULA
a(n) = A003266(n) mod A000045(n+1).
EXAMPLE
a(1) = 0 since A000045(1) = A000045(2) = 1 and 1 mod 1 = 0.
a(2) = (1 * 1) mod 2 = 1.
a(3) = (1 * 1 * 2) mod 3 = 2.
a(4) = (1 * 1 * 2 * 3) mod 5 = 1.
MATHEMATICA
a[n_] := Mod[Fibonorial[n], Fibonacci[n + 1]]; Array[a, 50] (* Amiram Eldar, Mar 29 2024 *)
PROG
(Python)
from sympy import fibonacci
def a(n):
a_n = 1
mod = fibonacci(n + 1)
for i in range(1, n + 1):
a_n = (a_n * fibonacci(i)) % mod
return a_n
(PARI) a(n) = my(f=fibonacci(n+1)); lift(prod(k=1, n, Mod(fibonacci(k), f))); \\ Michel Marcus, Apr 03 2024
CROSSREFS
Cf. A000045, A003266, A062347, A333599.
KEYWORD
nonn,new
AUTHOR
Adnan Baysal, Mar 29 2024
STATUS
approved
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Last modified April 25 07:07 EDT 2024. Contains 371964 sequences. (Running on oeis4.)