A363489 -id:A363489 - cp's OEIS frontend

A363948 Numbers whose prime indices have mean < 3/2.

2, 4, 8, 12, 16, 24, 32, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 432, 448, 480, 512, 576, 640, 768, 864, 896, 960, 1024, 1152, 1280, 1536, 1728, 1792, 1920, 2048, 2304, 2560, 2592, 2688, 2816, 2880, 3072, 3200, 3456, 3584, 3840, 4096, 4608

Offset: 1

Gus Wiseman, Jul 02 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The initial terms, prime indices, and means:
    2: {1} -> 1
    4: {1,1} -> 1
    8: {1,1,1} -> 1
   12: {1,1,2} -> 4/3
   16: {1,1,1,1} -> 1
   24: {1,1,1,2} -> 5/4
   32: {1,1,1,1,1} -> 1
   48: {1,1,1,1,2} -> 6/5
   64: {1,1,1,1,1,1} -> 1
   72: {1,1,1,2,2} -> 7/5
   80: {1,1,1,1,3} -> 7/5
   96: {1,1,1,1,1,2} -> 7/6

These partitions are counted by A363947.
Prime indices have mean A326567/A326568.
For low mode we have A360015, high A360013.
Positions of 1's in A363489.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A360005 gives twice the median of prime indices.
A363949 ranks partitions with low mean 1, counted by A025065.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124944, A327473, A327476, A327482, A359889, A363727, A363942, A363943, A363946, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]<3/2&]

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

Original entry on oeis.org

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}]

cp's OEIS Frontend

A363948 Numbers whose prime indices have mean < 3/2.

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