A363943 -id:A363943 - cp's OEIS frontend

A363950 Numbers whose prime indices have rounded-up mean 2.

3, 6, 9, 10, 12, 18, 20, 24, 27, 28, 30, 36, 40, 48, 54, 56, 60, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 264, 270, 280, 288, 300, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448

Offset: 1

Gus Wiseman, Jul 05 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 terms together with their prime indices begin:
     3: {2}
     6: {1,2}
     9: {2,2}
    10: {1,3}
    12: {1,1,2}
    18: {1,2,2}
    20: {1,1,3}
    24: {1,1,1,2}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    81: {2,2,2,2}

For mean 1 we have A000079 except 1.
Partitions of this type are counted by A026905 redoubled.
Equals the complement of A000079 in A344296.
Positions of 2's in A363944 (counted by column 2 of A363946).
For rounded mean 1 we have A363948, counted by A363947.
For rounded-down mean 1 we have A363949, counted by A025065.
The rounded-down or low version is A363954, counted by A363745.
A316413 ranks partitions with integer mean, counted by A067538.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
Cf. A327473, A327476, A359889, A360005, A360013, A360015, A363727, A363943.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Ceiling[Mean[prix[#]]]==2&]

A363488 Even numbers whose prime factorization has at least as many 2's as non-2's.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 84, 86, 88, 92, 94, 96, 100, 104, 106, 112, 116, 118, 120, 122, 124, 128, 132, 134, 136, 140, 142, 144, 146, 148, 152, 156, 158, 160

Offset: 1

Gus Wiseman, Jul 06 2023

nonn

The multiset of prime factors of n is row n of A027746.

Also numbers whose prime factors have low median 2, where the low median (see A124943) is either the middle part (for odd length), or the least of the two middle parts (for even length).

			The terms together with their prime indices begin:
     2: {1}            34: {1,7}             72: {1,1,1,2,2}
     4: {1,1}          36: {1,1,2,2}         74: {1,12}
     6: {1,2}          38: {1,8}             76: {1,1,8}
     8: {1,1,1}        40: {1,1,1,3}         80: {1,1,1,1,3}
    10: {1,3}          44: {1,1,5}           82: {1,13}
    12: {1,1,2}        46: {1,9}             84: {1,1,2,4}
    14: {1,4}          48: {1,1,1,1,2}       86: {1,14}
    16: {1,1,1,1}      52: {1,1,6}           88: {1,1,1,5}
    20: {1,1,3}        56: {1,1,1,4}         92: {1,1,9}
    22: {1,5}          58: {1,10}            94: {1,15}
    24: {1,1,1,2}      60: {1,1,2,3}         96: {1,1,1,1,1,2}
    26: {1,6}          62: {1,11}           100: {1,1,3,3}
    28: {1,1,4}        64: {1,1,1,1,1,1}    104: {1,1,1,6}
    32: {1,1,1,1,1}    68: {1,1,7}          106: {1,16}

Partitions of this type are counted by A027336.
The case without high median > 1 is A072978.
For mode instead of median we have A360015, high A360013.
Positions of 1's in A363941.
For mean instead of median we have A363949, high A000079.
The high version is A364056, positions of 1's in A363942.
A067538 counts partitions with integer mean, ranks A316413.
A112798 lists prime indices, length A001222, sum A056239.
A124943 counts partitions by low median, high A124944.
A363943 gives low mean of prime indices, triangle A363945.
Cf. A025065, A215366, A238495, A326567/A326568, A344296, A359889, A359908, A360005, A363486, A363487, A363944.

Select[Range[100],EvenQ[#]&&PrimeOmega[#]<=2*FactorInteger[#][[1,2]]&]

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

Original entry on oeis.org

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&]

A363947 Number of integer partitions of n with mean < 3/2.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 4, 4, 4, 7, 7, 7, 12, 12, 12, 19, 19, 19, 30, 30, 30, 45, 45, 45, 67, 67, 67, 97, 97, 97, 139, 139, 139, 195, 195, 195, 272, 272, 272, 373, 373, 373, 508, 508, 508, 684, 684, 684, 915, 915, 915, 1212, 1212, 1212, 1597, 1597, 1597, 2087

Offset: 0

Gus Wiseman, Jul 02 2023

nonn

			The partition y = (2,2,1) has mean 5/3, which is not less than 3/2, so y is not counted under 5.
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (11)  (111)  (211)   (2111)   (21111)   (22111)    (221111)
                    (1111)  (11111)  (111111)  (31111)    (311111)
                                               (211111)   (2111111)
                                               (1111111)  (11111111)

The high version is A000012 (all ones).
This is A000070 with each term repeated three times (see A025065 for two).
These partitions have ranks A363948.
The complement is counted by A364059.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A327482 counts partitions by integer mean.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A026905, A027336, A237984, A241131, A327472, A363724, A363745, A363943, A363949.

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

A363952 Number of integer partitions of n with low mode k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 4, 2, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 9, 3, 2, 0, 0, 0, 1, 0, 13, 5, 2, 1, 0, 0, 0, 1, 0, 18, 6, 3, 2, 0, 0, 0, 0, 1, 0, 26, 9, 3, 2, 1, 0, 0, 0, 0, 1, 0, 32, 13, 5, 3, 2, 0, 0, 0, 0, 0, 1, 0, 47, 16, 7, 3, 2, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jul 07 2023

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

Extending the terminology of A124943, the "low mode" of a multiset is the least mode.

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   0   1
   0   3   1   0   1
   0   4   2   0   0   1
   0   7   2   1   0   0   1
   0   9   3   2   0   0   0   1
   0  13   5   2   1   0   0   0   1
   0  18   6   3   2   0   0   0   0   1
   0  26   9   3   2   1   0   0   0   0   1
   0  32  13   5   3   2   0   0   0   0   0   1
   0  47  16   7   3   2   1   0   0   0   0   0   1
   0  60  21  10   4   3   2   0   0   0   0   0   0   1
   0  79  30  13   6   3   2   1   0   0   0   0   0   0   1
   0 104  38  17   7   4   3   2   0   0   0   0   0   0   0   1
Row n = 8 counts the following partitions:
  .  (71)        (62)     (53)   (44)  .  .  .  (8)
     (611)       (422)    (332)
     (521)       (3221)
     (5111)      (2222)
     (431)       (22211)
     (4211)
     (41111)
     (3311)
     (32111)
     (311111)
     (221111)
     (2111111)
     (11111111)

Row sums are A000041.
For median: A124943 (high A124944), rank statistic A363941 (high A363942).
Column k = 1 is A241131 (partitions w/ low mode 1), ranks A360015, A360013.
The rank statistic for this triangle is A363486.
For mean: A363945 (high A363946), rank statistic A363943 (high A363944).
The high version is A363953.
A008284 counts partitions by length, A058398 by mean.
A362612 counts partitions (max part) = (unique mode), ranks A362616.
A362614 counts partitions by number of modes, rank statistic A362611.
A362615 counts partitions by number of co-modes, rank statistic A362613.
Cf. A362610, A362608, A362607, A362609; A359178, A356862, A362605, A362606.
Cf. A002865, A025065, A026905, A067538, A237984, A363723, A363724, A363731.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], If[Length[#]==0,0,First[modes[#]]]==k&]],{n,0,15},{k,0,n}]

A363953 Number of integer partitions of n with high mode k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 2, 2, 1, 1, 1, 0, 7, 2, 1, 2, 1, 1, 1, 0, 9, 4, 2, 2, 2, 1, 1, 1, 0, 13, 6, 2, 2, 2, 2, 1, 1, 1, 0, 18, 7, 4, 3, 3, 2, 2, 1, 1, 1, 0, 26, 10, 5, 2, 3, 3, 2, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Jul 07 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.

Extending the terminology of A124944, the "high mode" in a multiset is the greatest mode.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  1  1  1
  0  3  1  1  1  1
  0  4  2  2  1  1  1
  0  7  2  1  2  1  1  1
  0  9  4  2  2  2  1  1  1
  0 13  6  2  2  2  2  1  1  1
  0 18  7  4  3  3  2  2  1  1  1
  0 26 10  5  2  3  3  2  2  1  1  1
  0 32 15  8  4  4  4  3  2  2  1  1  1
  0 47 19  9  5  3  4  4  3  2  2  1  1  1
  0 60 26 13  7  5  5  5  4  3  2  2  1  1  1
  0 79 34 18 10  6  5  5  5  4  3  2  2  1  1  1
Row n = 9 counts the following partitions:
  .  (711)        (522)     (333)   (441)  (54)   (63)   (72)  (81)  (9)
     (6111)       (4221)    (3321)  (432)  (531)  (621)
     (5211)       (3222)
     (51111)      (32211)
     (4311)       (22221)
     (42111)      (222111)
     (411111)
     (33111)
     (321111)
     (3111111)
     (2211111)
     (21111111)
     (111111111)

Row sums are A000041.
For median: A124944 (low A124943), rank statistic A363942 (low A363941).
Column k = 1 is A241131 (partitions w/ high mode 1), ranks A360013, A360015.
The rank statistic for this triangle is A363487, low A363486.
For mean: A363946 (low A363945), rank statistic A363944 (low A363943).
The low version is A363952.
A008284 counts partitions by length, A058398 by mean.
A362612 counts partitions (max part) = (unique mode), ranks A362616.
A362614 counts partitions by number of modes, rank statistic A362611.
A362615 counts partitions by number of co-modes, rank statistic A362613.
Cf. A362610, A362608, A362607, A362609; A359178, A356862, A362605, A362606.
Cf. A002865, A025065, A026905, A237984, A363723, A363724, A363731.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], If[Length[#]==0,0,Last[modes[#]]]==k&]],{n,0,15},{k,0,n}]

A363745 Number of integer partitions of n whose rounded-down mean is 2.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 4, 10, 6, 16, 21, 24, 32, 58, 47, 85, 111, 119, 158, 248, 217, 341, 442, 461, 596, 867, 792, 1151, 1465, 1506, 1916, 2652, 2477, 3423, 4298, 4381, 5488, 7334, 6956, 9280, 11503, 11663, 14429, 18781, 17992, 23383, 28675, 28970, 35449, 45203

Offset: 0

Gus Wiseman, Jul 05 2023

nonn

			The a(2) = 1 through a(10) = 16 partitions:
  (2)  .  (22)  (32)  (222)  (322)  (332)   (3222)  (3322)
          (31)  (41)  (321)  (331)  (422)   (3321)  (3331)
                      (411)  (421)  (431)   (4221)  (4222)
                             (511)  (521)   (4311)  (4321)
                                    (611)   (5211)  (4411)
                                    (2222)  (6111)  (5221)
                                    (3221)          (5311)
                                    (3311)          (6211)
                                    (4211)          (7111)
                                    (5111)          (22222)
                                                    (32221)
                                                    (33211)
                                                    (42211)
                                                    (43111)
                                                    (52111)
                                                    (61111)

For 1 instead of 2 we have A025065, ranks A363949.
The high version is A026905 reduplicated, ranks A363950.
Column k = 2 of A363945.
These partitions have ranks A363954.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A027336, A237984, A241131, A327472, A327482, A363723, A363943, A363944, A363946.

Table[Length[Select[IntegerPartitions[n],Floor[Mean[#]]==2&]],{n,0,30}]

A126594 Floor of the average of the prime factors of n with multiplicity.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 4, 4, 2, 17, 2, 19, 3, 5, 6, 23, 2, 5, 7, 3, 3, 29, 3, 31, 2, 7, 9, 6, 2, 37, 10, 8, 2, 41, 4, 43, 5, 3, 12, 47, 2, 7, 4, 10, 5, 53, 2, 8, 3, 11, 15, 59, 3, 61, 16, 4, 2, 9, 5, 67, 7, 13, 4, 71, 2, 73, 19, 4, 7, 9, 6, 79, 2, 3, 21, 83, 3, 11, 22, 16, 4, 89, 3, 10

Offset: 2

Cino Hilliard, Jan 06 2007

Cf. A067629 (rounding instead of flooring), A076690.
This is the floor of A123528/A123529.
Without multiplicity we have A363895.
For prime indices instead of factors we have A363943, triangle A363945.
Positions of first appearances are A364037.
The ceiling is A364156.
Positions of 2's are A364157, for prime indices A363949.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, ranks A316413.
A078175 lists numbers with integer mean of prime factors.
Cf. A026905, A326567/A326568, A327473, A327476, A363944, A363948, A363950.

Table[Floor[(Plus@@Times@@@FactorInteger[n])/PrimeOmega[n]], {n, 2, 90}] (* Alonso del Arte, May 21 2012 *)

avg(n) = { local(x,j,ln) for(x=2,n,a=ifactor(x); ln=length(a); print1(floor(sum(j=1,ln,a[j])/ln)",")) } ifactor(n) = \The vector of the prime factors of n with multiplicity. { local(f,j,k,flist); flist=[]; f=Vec(factor(n)); for(j=1,length(f[1]), for(k = 1,f[2][j],flist = concat(flist,f[1][j]) ); ); return(flist) }

a(p^n)=p, p prime, n >= 1. - Philippe Deléham, Nov 23 2008

a(n) = floor(A001414(n)/A001222(n)). - Philippe Deléham, Nov 24 2008

A363954 Numbers whose prime indices have low mean 2.

Original entry on oeis.org

3, 9, 10, 14, 15, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 70, 75, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 189, 196, 198, 204, 208, 210, 220, 225, 234, 243, 250, 252, 260, 264, 270, 272, 280, 294, 297, 300, 304, 308, 312, 315, 330, 350

Offset: 1

Gus Wiseman, Jul 05 2023

nonn

Extending the terminology of A124944, the "low mean" of a multiset is obtained by taking the mean and rounding down.

			The terms together with their prime indices begin:
     3: {2}
     9: {2,2}
    10: {1,3}
    14: {1,4}
    15: {2,3}
    27: {2,2,2}
    28: {1,1,4}
    30: {1,2,3}
    42: {1,2,4}
    44: {1,1,5}
    45: {2,2,3}
    50: {1,3,3}
    52: {1,1,6}
    63: {2,2,4}
    66: {1,2,5}
    70: {1,3,4}
    75: {2,3,3}
    81: {2,2,2,2}
    84: {1,1,2,4}
    88: {1,1,1,5}
    90: {1,2,2,3}
   100: {1,1,3,3}

Partitions of this type are counted by A363745.
Positions of 2's in A363943 (high A363944), triangle A363945 (high A363946).
For mean 1 we have A363949.
The high version is A363950, counted by A026905.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
A363941 gives low median of prime indices, triangle A124943.
A363942 gives high median of prime indices, triangle A124944.
A363948 lists numbers whose prime indices have mean 1, counted by A363947.
Cf. A327473, A327476, A359889, A360013, A360015, A363488, A363951.

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

A363489 Rounded mean of the multiset of prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 1, 6, 2, 2, 1, 7, 2, 8, 2, 3, 3, 9, 1, 3, 4, 2, 2, 10, 2, 11, 1, 4, 4, 4, 2, 12, 4, 4, 2, 13, 2, 14, 2, 2, 5, 15, 1, 4, 2, 4, 3, 16, 2, 4, 2, 5, 6, 17, 2, 18, 6, 3, 1, 4, 3, 19, 3, 6, 3, 20, 1, 21, 6, 3, 3, 4, 3, 22, 1, 2, 7

Offset: 1

Gus Wiseman, Jul 07 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.

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

			The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, which rounds to 2, so a(180) = 2.

Wikipedia, Rounding.

Positions of first appearances are 1 and A000040.
Before rounding we had A326567/A326568.
For rounded-down: A363943, triangle A363945.
For rounded-up: A363944, triangle A363946.
Positions of 1's are A363948, complement A364059.
The triangle for this statistic (rounded mean) is A364060.
For prime factors instead of indices we have A364061.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
Cf. A327473, A327476, A327482, A359889, A360005, A363947, A363949, A363951.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Round[Mean[prix[n]]]],{n,100}]

cp's OEIS Frontend

A363950 Numbers whose prime indices have rounded-up mean 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363488 Even numbers whose prime factorization has at least as many 2's as non-2's.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363947 Number of integer partitions of n with mean < 3/2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A363952 Number of integer partitions of n with low mode k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363953 Number of integer partitions of n with high mode k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A363745 Number of integer partitions of n whose rounded-down mean is 2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A126594 Floor of the average of the prime factors of n with multiplicity.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A363954 Numbers whose prime indices have low mean 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica