A327471
Number of subsets of {1..n} not containing their mean.
Original entry on oeis.org
1, 1, 2, 4, 10, 22, 48, 102, 214, 440, 900, 1830, 3706, 7486, 15092, 30380, 61100, 122780, 246566, 494912, 992984, 1991620, 3993446, 8005388, 16044460, 32150584, 64414460, 129037790, 258462026, 517641086, 1036616262, 2075721252, 4156096036, 8320912744, 16658202200
Offset: 0
The a(1) = 1 through a(5) = 22 subsets:
{} {} {} {} {}
{1,2} {1,2} {1,2} {1,2}
{1,3} {1,3} {1,3}
{2,3} {1,4} {1,4}
{2,3} {1,5}
{2,4} {2,3}
{3,4} {2,4}
{1,2,4} {2,5}
{1,3,4} {3,4}
{1,2,3,4} {3,5}
{4,5}
{1,2,4}
{1,2,5}
{1,3,4}
{1,4,5}
{2,3,5}
{2,4,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
Subsets containing their mean are
A065795.
Subsets containing n but not their mean are
A327477.
Partitions not containing their mean are
A327472.
Strict partitions not containing their mean are
A240851.
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Table[Length[Select[Subsets[Range[n]],!MemberQ[#,Mean[#]]&]],{n,0,10}]
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from sympy import totient, divisors
def A327471(n): return (1<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))>>1) # Chai Wah Wu, Feb 22 2023
A327474
Number of distinct means of subsets of {1..n}, where {} has mean 0.
Original entry on oeis.org
1, 2, 4, 6, 10, 16, 26, 38, 56, 78, 106, 138, 180, 226, 284, 348, 420, 500, 596, 698, 818, 946, 1086, 1236, 1408, 1588, 1788, 2000, 2230, 2472, 2742, 3020, 3328, 3652, 3996, 4356, 4740, 5136, 5568, 6018, 6492, 6982, 7512, 8054, 8638, 9242, 9870, 10520, 11216
Offset: 0
The a(3) = 6 distinct means are 0, 1, 3/2, 2, 5/2, 3.
The version for only nonempty subsets is
A135342.
-
a:= proc(n) option remember; `if`(n<4, [1, 2, 4, 6][n+1],
2*a(n-1)-a(n-2)+numtheory[phi](n-1))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Feb 22 2023
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Table[Length[Union[Mean/@Subsets[Range[n]]]],{n,0,10}]
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from itertools import count, islice
from sympy import totient
def A327474_gen(): # generator of terms
a, b = 4, 6
yield from (1,2,4,6)
for n in count(3):
a, b = b, (b<<1)-a+totient(n)
yield b
A327474_list = list(islice(A327474_gen(),30)) # Chai Wah Wu, Feb 22 2023
A327477
Number of subsets of {1..n} containing n whose mean is not an element.
Original entry on oeis.org
0, 0, 1, 2, 6, 12, 26, 54, 112, 226, 460, 930, 1876, 3780, 7606, 15288, 30720, 61680, 123786, 248346, 498072, 998636, 2001826, 4011942, 8039072, 16106124, 32263876, 64623330, 129424236, 259179060, 518975176, 1039104990, 2080374784, 4164816708, 8337289456
Offset: 0
The a(1) = 1 through a(5) = 12 subsets:
{1,2} {1,3} {1,4} {1,5}
{2,3} {2,4} {2,5}
{3,4} {3,5}
{1,2,4} {4,5}
{1,3,4} {1,2,5}
{1,2,3,4} {1,4,5}
{2,3,5}
{2,4,5}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
Subsets whose mean is an element are
A065795.
Subsets whose mean is not an element are
A327471.
Subsets containing n whose mean is an element are
A000016.
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Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!MemberQ[#,Mean[#]]&]],{n,0,10}]
-
from sympy import totient, divisors
def A327477(n): return (1<>(~n&n-1).bit_length(),generator=True))//n if n else 0 # Chai Wah Wu, Feb 21 2023
A327478
Numbers whose average binary index is also a binary index.
Original entry on oeis.org
1, 2, 4, 7, 8, 14, 16, 21, 28, 31, 32, 39, 42, 56, 57, 62, 64, 73, 78, 84, 93, 107, 112, 114, 124, 127, 128, 141, 146, 155, 156, 168, 175, 177, 186, 214, 217, 224, 228, 245, 248, 254, 256, 267, 273, 282, 287, 292, 310, 312, 313, 336, 341, 350, 354, 371, 372
Offset: 1
The sequence of terms together with their binary indices begins:
1: 1
2: 2
4: 3
7: 1 2 3
8: 4
14: 2 3 4
16: 5
21: 1 3 5
28: 3 4 5
31: 1 2 3 4 5
32: 6
39: 1 2 3 6
42: 2 4 6
56: 4 5 6
57: 1 4 5 6
61: 2 3 4 5 6
Numbers whose binary indices have integer mean are
A326669.
Cf.
A000016,
A000120,
A029931,
A048793,
A065795,
A070939,
A237984,
A240850,
A327473,
A327474,
A327481.
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],MemberQ[bpe[#],Mean[bpe[#]]]&]
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