login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: keyword:new
Displaying 1-10 of 422 results found. page 1 2 3 4 5 6 7 8 9 10 ... 43
     Sort: relevance | references | number | modified | created      Format: long | short | data
A371051 Ternary numbers consisting of a run of 1's, then a run of 2's, then a run of 0's. +0
0
120, 1120, 1200, 1220, 11120, 11200, 11220, 12000, 12200, 12220, 111120, 111200, 111220, 112000, 112200, 112220, 120000, 122000, 122200, 122220, 1111120, 1111200, 1111220, 1112000, 1112200, 1112220, 1120000, 1122000, 1122200, 1122220, 1200000, 1220000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
MATHEMATICA
Map[#[[1]] &, Select[Map[{#, Map[#[[1]] &, Split[IntegerDigits[#, 3]]] == {1, 2, 0}} &,
Range[0, 4000, 3]], #[[2]] &]] (* A371050 *)
ToExpression[Map[IntegerString[#, 3] &, %]] (* this sequence *)
(* Peter J. C. Moses, Mar 06 2024 *)
CROSSREFS
KEYWORD
nonn,base,new
AUTHOR
Clark Kimberling, Mar 17 2024
STATUS
approved
A371050 Numbers whose ternary representation consists of a run of 1's, then a run of 2's, then a run of 0's. +0
0
15, 42, 45, 51, 123, 126, 132, 135, 153, 159, 366, 369, 375, 378, 396, 402, 405, 459, 477, 483, 1095, 1098, 1104, 1107, 1125, 1131, 1134, 1188, 1206, 1212, 1215, 1377, 1431, 1449, 1455, 3282, 3285, 3291, 3294, 3312, 3318, 3321, 3375, 3393, 3399, 3402, 3564 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All the numbers are multiples of 3.
LINKS
EXAMPLE
The ternary represerntations of 15, 42, 45 are 120, 1120, 1200.
MATHEMATICA
Map[#[[1]] &, Select[Map[{#, Map[#[[1]] &, Split[IntegerDigits[#, 3]]] == {1, 2, 0}} &,
Range[0, 4000, 3]], #[[2]] &]] (* this sequence *)
Map[IntegerString[#, 3] &, %] (* A371051 *)
(* Peter J. C. Moses, Mar 06 2024 *)
CROSSREFS
KEYWORD
nonn,base,new
AUTHOR
Clark Kimberling, Mar 17 2024
STATUS
approved
A371052 Numbers whose ternary representation consists of a run of 2's, then a run of 1's, then a run of 0's. +0
0
21, 63, 66, 75, 189, 198, 201, 225, 228, 237, 567, 594, 603, 606, 675, 684, 687, 711, 714, 723, 1701, 1782, 1809, 1818, 1821, 2025, 2052, 2061, 2064, 2133, 2142, 2145, 2169, 2172, 2181, 5103, 5346, 5427, 5454, 5463, 5466 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All the numbers are multiples of 3.
LINKS
EXAMPLE
The ternary representations of 21, 63, 66 are 210, 2100, 2110.
MATHEMATICA
Map[#[[1]] &, Select[Map[{#, Map[#[[1]] &, Split[IntegerDigits[#, 3]]] == {2, 1, 0}} &,
Range[0, 6000, 3]], #[[2]] &]] (* this sequence *)
ToExpression[Map[IntegerString[#, 3] &, %]] (* A371053 *)
(* Peter J. C. Moses, Mar 06 2024 *)
CROSSREFS
KEYWORD
nonn,base,new
AUTHOR
Clark Kimberling, Mar 17 2024
STATUS
approved
A371281 Last digit of the product of decimal digits of n. +0
0
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 0, 6, 2, 8, 4, 0, 6, 2, 8, 4, 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 0, 9, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
n=0 is taken as one 0 digit so that its product of digits is A007954(0) = 0.
LINKS
FORMULA
a(n) = A007954(n) mod 10.
a(n) = A010879(A007954(n)).
EXAMPLE
n = 15: a(15) = 1*5 mod 10 = 5.
n = 26: a(26) = 2*6 mod 10 = 2.
MAPLE
a:= n-> `if`(n<10, n, irem(n, 10, 'q')*a(q) mod 10):
seq(a(n), n=0..92); # Alois P. Heinz, Mar 17 2024
MATHEMATICA
a[n_] := Mod[Times @@ IntegerDigits[n], 10]; Array[a, 100, 0] (* Amiram Eldar, Mar 17 2024 *)
PROG
(Python)
from math import prod
def a(n): return prod(map(int, str(n)))%10
print([a(n) for n in range(93)]) # Michael S. Branicky, Mar 17 2024
(PARI) a(n) = if (n==0, 0, vecprod(digits(n)) % 10); \\ Michel Marcus, Mar 17 2024
CROSSREFS
Cf. A007954, A010879, A036987 (similar for 2 instead of 10), A053837.
KEYWORD
nonn,base,new
AUTHOR
Ctibor O. Zizka, Mar 17 2024
STATUS
approved
A371278 Numbers k > 1 such that k / A054841(k) is an integer. +0
0
2, 4, 12, 16, 144, 192, 256, 1728, 3888, 4320, 6480, 7200, 11520, 13122, 14580, 15360, 20736, 36864, 49152, 65536, 107520, 344064, 384000, 589824, 691200, 1244160, 1259712, 1327104, 2211840, 2304960, 2963520, 2985984, 3932160, 3981312, 4478976, 4500000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
EXAMPLE
k = 144: A054841(144) = 24 because 144 = 3^2 * 2^4, 144/A054841(144) = 144/24 = 6, thus 144 is a term.
MATHEMATICA
q[n_] := Module[{f = FactorInteger[n]}, Divisible[n, Total[10^(PrimePi[f[[;; , 1]]] - 1) * f[[;; , 2]]]]]; q[1] = False; Select[Range[10^5], q] (* Amiram Eldar, Mar 17 2024 *)
PROG
(Python)
from sympy import factorint, primepi
def ok(n): return n > 1 and n%sum(e*10**(primepi(p)-1) for p, e in factorint(n).items()) == 0
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Mar 17 2024
CROSSREFS
Cf. A054841.
KEYWORD
nonn,new
AUTHOR
Ctibor O. Zizka, Mar 17 2024
EXTENSIONS
More terms from Michel Marcus, Mar 17 2024
STATUS
approved
A371059 Number of conjugacy classes of pairs of commuting elements in the alternating group A_n. +0
0
1, 1, 9, 14, 22, 44, 74, 160, 256, 462, 817, 1494, 2543, 4427, 7699, 13352, 22616, 38610, 65052, 110004, 182961, 305007, 503299, 830648, 1356227, 2212790, 3583419, 5790836 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The number of conjugacy classes of pairs of commuting elements in a finite group G is the cardinality of the set {c(a,b) | a,b in G and ab=ba} where c(a,b) = {(gag^(-1),gbg^(-1)) | g in G}.
It is equal to the number of conjugacy classes within the centralizers of class representatives of G.
This reformulation was employed in the sequence-generating program.
It is also equal to the rank of the modular fusion category Z(Rep(G)), the Drinfeld center of Rep(G).
These reformulations are explained in the linked MathOverflow posts.
REFERENCES
A. Davydov, Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds. J. Math. Phys. 55 (2014), no. 9, 092305, 13 pp.
LINKS
PROG
(GAP)
List([1..10], n->Sum(List(ConjugacyClasses(AlternatingGroup(n)), c->NrConjugacyClasses(Centralizer(AlternatingGroup(n), Representative(c))))));
CROSSREFS
Cf. A000702.
KEYWORD
nonn,more,new
AUTHOR
Sébastien Palcoux, Mar 11 2024
STATUS
approved
A371032 a(n) = binary string starting with 1 such that the runs of identical bits in a(n) have lengths n, n-1, n-2, ..., 3, 2, 1. +0
1
1, 110, 111001, 1111000110, 111110000111001, 111111000001111000110, 1111111000000111110000111001, 111111110000000111111000001111000110, 111111111000000001111111000000111110000111001, 1111111111000000000111111110000000111111000001111000110 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
EXAMPLE
a(1) = 1 has runlength 1; a(2) = 110 has runlengths 2,1; a(3) = 111001 has runlengths 3,2,1.
MATHEMATICA
Flatten[Table[Flatten[Map[ConstantArray[Mod[#, 2], n + 1 - #] &, Range[n]]], {n, 10}]] (* Peter J. C. Moses, Mar 08 2024 *)
PROG
(Python)
def A371032(n):
c = 0
for i in range(n):
c = (m:=10**(n-i))*c
if i&1^1:
c += (m-1)//9
return c # Chai Wah Wu, Mar 18 2024
CROSSREFS
Cf. A007088, A371033 (decimal version).
KEYWORD
nonn,base,new
AUTHOR
Clark Kimberling, Mar 09 2024
STATUS
approved
A371296 E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x)^2)). +0
0
1, 2, 26, 674, 26682, 1429682, 96867178, 7946279490, 765861255002, 84837503946962, 10621798904563530, 1483378875680954210, 228626616449674796602, 38549099486166110798322, 7058696888173770772536362, 1394913467379909728350803074, 295904373562519633314958421274 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
FORMULA
a(n) = (1/(2*n+1)!) * Sum_{k=0..n} 2^k * (2*n+k)! * Stirling2(n,k).
PROG
(PARI) a(n) = sum(k=0, n, 2^k*(2*n+k)!*stirling(n, k, 2))/(2*n+1)!;
CROSSREFS
Cf. A367134.
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Mar 18 2024
STATUS
approved
A370882 Square array T(n,k) = 9*2^k - n read by ascending antidiagonals. +0
0
9, 8, 18, 7, 17, 36, 6, 16, 35, 72, 5, 15, 34, 71, 144, 4, 14, 33, 70, 143, 288, 3, 13, 32, 69, 142, 287, 576, 2, 12, 31, 68, 141, 286, 575, 1152, 1, 11, 30, 67, 140, 285, 574, 1151, 2304, 0, 10, 29, 66, 139, 284, 573, 1150, 2303, 4608, -1, 9, 28, 65, 138, 283, 572, 1149, 2302, 4607, 9216 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Just after A367559 and A368826.
LINKS
FORMULA
T(0,k) = 9*2^k = A005010(k);
T(1,k) = 9*2^k - 1 = A052996(k+2);
T(2,k) = 9*2^k - 2 = A176449(k);
T(3,k) = 9*2^k - 3 = 3*A083329(k);
T(4,k) = 9*2^k - 4 = A053209(k);
T(5,k) = 9*2^k - 5 = A304383(k+3);
T(6,k) = 9*2^k - 6 = 3*A033484(k);
T(7,k) = 9*2^k - 7 = A154251(k+1);
T(8,k) = 9*2^k - 8 = A048491(k);
T(9,k) = 9*2^k - 9 = 3*A000225(k).
G.f.: (9 - 9*y + x*(11*y - 10))/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Mar 17 2024
EXAMPLE
Table begins:
k=0 1 2 3 4 5
n=0: 9 18 36 72 144 288 ...
n=1: 8 17 35 71 143 287 ...
n=2: 7 16 34 70 142 286 ...
n=3: 6 15 33 69 141 285 ...
n=4: 5 14 32 68 140 284 ...
n=5: 4 13 31 67 139 283 ...
Every line has the signature (3,-2). For n=1: 3*17 - 2*8 = 35.
Main diagonal's difference table:
9 17 34 69 140 283 570 1145 ... = b(n)
8 17 35 71 143 287 575 1151 ... = A052996(n+2)
9 18 36 72 144 288 576 1152 ... = A005010(n)
...
b(n+1) - 2*b(n) = A023443(n).
MATHEMATICA
T[n_, k_] := 9*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)
CROSSREFS
Cf. A023443.
KEYWORD
sign,tabl,new
AUTHOR
Paul Curtz, Mar 05 2024
STATUS
approved
A371178 Number of integer partitions of n containing all divisors of all parts. +0
0
1, 1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 37, 48, 62, 80, 101, 127, 162, 202, 252, 312, 386, 475, 585, 713, 869, 1056, 1278, 1541, 1859, 2232, 2675, 3196, 3811, 4534, 5386, 6379, 7547, 8908, 10497, 12345, 14501, 16999, 19897, 23253, 27135, 31618, 36796, 42756 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
The Heinz numbers of these partitions are given by A371177.
Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.
LINKS
EXAMPLE
The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - David A. Corneth, Mar 18 2024
The a(0) = 1 through a(8) = 12 partitions:
() (1) (11) (21) (31) (221) (51) (331) (71)
(111) (211) (311) (321) (421) (521)
(1111) (2111) (2211) (511) (3221)
(11111) (3111) (2221) (3311)
(21111) (3211) (4211)
(111111) (22111) (5111)
(31111) (22211)
(211111) (32111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], SubsetQ[#, Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
The LHS is represented by A001221, distinct case of A001222.
For partitions with no divisors of parts we have A305148, ranks A316476.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
Counting all parts on the LHS gives A371130, ranks A370802.
The complement is counted by A371132.
For submultisets instead of distinct parts we have A371172, ranks A371165.
These partitions have ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
KEYWORD
nonn,new
AUTHOR
Gus Wiseman, Mar 17 2024
STATUS
approved
page 1 2 3 4 5 6 7 8 9 10 ... 43

Search completed in 0.119 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)