Search: keyword:new
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A371051
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Ternary numbers consisting of a run of 1's, then a run of 2's, then a run of 0's.
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+0
0
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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
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OFFSET
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1,1
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LINKS
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MATHEMATICA
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Map[#[[1]] &, Select[Map[{#, Map[#[[1]] &, Split[IntegerDigits[#, 3]]] == {1, 2, 0}} &,
Range[0, 4000, 3]], #[[2]] &]] (* A371050 *)
ToExpression[Map[IntegerString[#, 3] &, %]] (* this sequence *)
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CROSSREFS
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KEYWORD
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nonn,base,new
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AUTHOR
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STATUS
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approved
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A371050
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Numbers whose ternary representation consists of a run of 1's, then a run of 2's, then a run of 0's.
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+0
0
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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
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OFFSET
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1,1
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COMMENTS
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All the numbers are multiples of 3.
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LINKS
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EXAMPLE
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The ternary represerntations of 15, 42, 45 are 120, 1120, 1200.
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MATHEMATICA
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Map[#[[1]] &, Select[Map[{#, Map[#[[1]] &, Split[IntegerDigits[#, 3]]] == {1, 2, 0}} &,
Range[0, 4000, 3]], #[[2]] &]] (* this sequence *)
Map[IntegerString[#, 3] &, %] (* A371051 *)
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CROSSREFS
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KEYWORD
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nonn,base,new
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AUTHOR
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STATUS
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approved
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A371052
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Numbers whose ternary representation consists of a run of 2's, then a run of 1's, then a run of 0's.
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+0
0
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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
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OFFSET
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1,1
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COMMENTS
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All the numbers are multiples of 3.
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LINKS
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EXAMPLE
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The ternary representations of 21, 63, 66 are 210, 2100, 2110.
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MATHEMATICA
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Map[#[[1]] &, Select[Map[{#, Map[#[[1]] &, Split[IntegerDigits[#, 3]]] == {2, 1, 0}} &,
Range[0, 6000, 3]], #[[2]] &]] (* this sequence *)
ToExpression[Map[IntegerString[#, 3] &, %]] (* A371053 *)
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CROSSREFS
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KEYWORD
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nonn,base,new
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AUTHOR
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STATUS
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approved
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A371281
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Last digit of the product of decimal digits of n.
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+0
0
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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
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OFFSET
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0,3
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COMMENTS
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n=0 is taken as one 0 digit so that its product of digits is A007954(0) = 0.
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LINKS
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FORMULA
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EXAMPLE
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n = 15: a(15) = 1*5 mod 10 = 5.
n = 26: a(26) = 2*6 mod 10 = 2.
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MAPLE
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a:= n-> `if`(n<10, n, irem(n, 10, 'q')*a(q) mod 10):
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MATHEMATICA
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a[n_] := Mod[Times @@ IntegerDigits[n], 10]; Array[a, 100, 0] (* Amiram Eldar, Mar 17 2024 *)
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PROG
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(Python)
from math import prod
def a(n): return prod(map(int, str(n)))%10
(PARI) a(n) = if (n==0, 0, vecprod(digits(n)) % 10); \\ Michel Marcus, Mar 17 2024
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CROSSREFS
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KEYWORD
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nonn,base,new
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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k = 144: A054841(144) = 24 because 144 = 3^2 * 2^4, 144/A054841(144) = 144/24 = 6, thus 144 is a term.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A371059
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Number of conjugacy classes of pairs of commuting elements in the alternating group A_n.
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+0
0
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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
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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A. Davydov, Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds. J. Math. Phys. 55 (2014), no. 9, 092305, 13 pp.
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LINKS
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PROG
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(GAP)
List([1..10], n->Sum(List(ConjugacyClasses(AlternatingGroup(n)), c->NrConjugacyClasses(Centralizer(AlternatingGroup(n), Representative(c))))));
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CROSSREFS
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KEYWORD
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nonn,more,new
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AUTHOR
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STATUS
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approved
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A371032
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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.
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+0
1
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1, 110, 111001, 1111000110, 111110000111001, 111111000001111000110, 1111111000000111110000111001, 111111110000000111111000001111000110, 111111111000000001111111000000111110000111001, 1111111111000000000111111110000000111111000001111000110
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OFFSET
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1,2
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LINKS
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EXAMPLE
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a(1) = 1 has runlength 1; a(2) = 110 has runlengths 2,1; a(3) = 111001 has runlengths 3,2,1.
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MATHEMATICA
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Flatten[Table[Flatten[Map[ConstantArray[Mod[#, 2], n + 1 - #] &, Range[n]]], {n, 10}]] (* Peter J. C. Moses, Mar 08 2024 *)
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PROG
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(Python)
c = 0
for i in range(n):
c = (m:=10**(n-i))*c
if i&1^1:
c += (m-1)//9
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CROSSREFS
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KEYWORD
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nonn,base,new
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AUTHOR
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STATUS
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approved
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A371296
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E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x)^2)).
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+0
0
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1, 2, 26, 674, 26682, 1429682, 96867178, 7946279490, 765861255002, 84837503946962, 10621798904563530, 1483378875680954210, 228626616449674796602, 38549099486166110798322, 7058696888173770772536362, 1394913467379909728350803074, 295904373562519633314958421274
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (1/(2*n+1)!) * Sum_{k=0..n} 2^k * (2*n+k)! * Stirling2(n,k).
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PROG
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(PARI) a(n) = sum(k=0, n, 2^k*(2*n+k)!*stirling(n, k, 2))/(2*n+1)!;
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CROSSREFS
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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A370882
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Square array T(n,k) = 9*2^k - n read by ascending antidiagonals.
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+0
0
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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
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OFFSET
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0,1
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COMMENTS
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LINKS
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FORMULA
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G.f.: (9 - 9*y + x*(11*y - 10))/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Mar 17 2024
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EXAMPLE
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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)
...
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MATHEMATICA
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T[n_, k_] := 9*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)
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CROSSREFS
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Cf. A000225, A033484, A048491, A005010, A052996, A053209, A083329, A154251, A176449, A304383, A367559, A368826.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A371178
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Number of integer partitions of n containing all divisors of all parts.
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+0
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], SubsetQ[#, Union@@Divisors/@#]&]], {n, 0, 30}]
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CROSSREFS
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For partitions with no divisors of parts we have A305148, ranks A316476.
The complement is counted by A371132.
For submultisets instead of distinct parts we have A371172, ranks A371165.
These partitions have ranks A371177.
A008284 counts partitions by length.
Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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