A363943 -id:A363943 - cp's OEIS frontend

A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1, 0, 2, 4, 0, 0, 1, 0, 2, 5, 3, 0, 0, 1, 0, 4, 7, 0, 3, 0, 0, 1, 0, 4, 8, 5, 4, 0, 0, 0, 1, 0, 4, 14, 7, 4, 0, 0, 0, 0, 1, 0, 7, 21, 8, 0, 5, 0, 0, 0, 0, 1, 0, 7, 22, 11, 10, 0, 5, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jul 07 2023

We use the "rounding half to even" rule, see link.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  2  0  1
  0  2  4  0  0  1
  0  2  5  3  0  0  1
  0  4  7  0  3  0  0  1
  0  4  8  5  4  0  0  0  1
  0  4 14  7  4  0  0  0  0  1
  0  7 21  8  0  5  0  0  0  0  1
  0  7 22 11 10  0  5  0  0  0  0  1
  0  7 36 15 12  0  6  0  0  0  0  0  1
  0 12 32 36 14  0  6  0  0  0  0  0  0  1
  0 12 53 23 23 16  0  7  0  0  0  0  0  0  1
  0 12 80 30 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (31111)    (511)   .  (61)  .  .  (7)
     (22111)    (421)      (52)
     (211111)   (4111)     (43)
     (1111111)  (331)
                (322)
                (3211)
                (2221)

Wikipedia, Rounding.

Row sums are A000041.
The rank statistic for this triangle is A363489.
The version for low mean is A363945, rank statistic A363943.
The version for high mean is A363946, rank statistic A363944.
Column k = 1 is A363947 (A026905 tripled).
A008284 counts partitions by length, A058398 by mean.
A026905 redoubled counts partitions with high mean 2, ranks A363950.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
More triangles: A124943, A124944, A363952, A363953.
Cf. A002865, A025065, A237984, A327472, A327482, A363723, A363724, A363731.

Table[If[n==k==0,1,Length[Select[IntegerPartitions[n], Round[Mean[#]]==k&]]],{n,0,15},{k,0,n}]

A364156 Ceiling of the mean of the prime factors of n (with multiplicity).

Original entry on oeis.org

0, 2, 3, 2, 5, 3, 7, 2, 3, 4, 11, 3, 13, 5, 4, 2, 17, 3, 19, 3, 5, 7, 23, 3, 5, 8, 3, 4, 29, 4, 31, 2, 7, 10, 6, 3, 37, 11, 8, 3, 41, 4, 43, 5, 4, 13, 47, 3, 7, 4, 10, 6, 53, 3, 8, 4, 11, 16, 59, 3, 61, 17, 5, 2, 9, 6, 67, 7, 13, 5, 71, 3, 73, 20, 5, 8, 9, 6

Offset: 1

Gus Wiseman, Jul 18 2023

nonn

			The prime factors of 450 are {2,3,3,5,5}, with mean 18/5, so a(450) = 4.

For median of prime indices we have triangle A124944, low A124943.
The round version is A067629.
The floor version is A126594.
A027746 lists prime factors, indices A112798.
A078175 lists numbers with integer mean of prime factors.
A123528/A123529 gives mean of prime factors, A326567/A326568 prime indices.
Cf. A026905, A051293, A316413, A327473, A327476, A327482, A363895, A363943, A363948, A363950, A364037.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[If[n==1,0,Ceiling[Mean[prifacs[n]]]],{n,100}]

Ceiling of A123528(n)/A123529(n).

A364157 Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 162, 192, 216, 224, 240, 256, 288, 320, 324, 360, 384, 432, 448, 480, 486, 512, 576, 640, 648, 672, 720, 768, 800, 864, 896, 960, 972, 1024, 1080, 1152, 1280, 1296, 1344

Offset: 1

Gus Wiseman, Jul 18 2023

nonn

			The terms together with their prime factors begin:
   2 = 2
   4 = 2*2
   6 = 2*3
   8 = 2*2*2
  12 = 2*2*3
  16 = 2*2*2*2
  18 = 2*3*3
  24 = 2*2*2*3
  32 = 2*2*2*2*2
  36 = 2*2*3*3
  40 = 2*2*2*5
  48 = 2*2*2*2*3
  54 = 2*3*3*3
  64 = 2*2*2*2*2*2
  72 = 2*2*2*3*3
  80 = 2*2*2*2*5
  96 = 2*2*2*2*2*3

Without multiplicity we appear to have A007694.
Prime factors are listed by A027746, indices A112798.
Positions of 2's in A126594, positions of first appearances A364037.
For prime indices and ceiling we have A363950, counted by A026905.
For prime indices we have A363954 (or A363949), counted by A363745.
A078175 lists numbers with integer mean of prime factors.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A316413 ranks partitions with integer mean, counted by A067538.
A363895 gives floor of mean of distinct prime factors.
A363943 gives floor of mean of prime indices, ceiling A363944.
Cf. A001222, A056239, A327473, A327476, A360013, A360015, A363488, A363945.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Floor[Mean[prifacs[#]]]==2&]

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A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.

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A364156 Ceiling of the mean of the prime factors of n (with multiplicity).

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A364157 Numbers whose rounded-down (floor) mean of prime factors (with multiplicity) is 2.

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