A359908 -id:A359908 - cp's OEIS frontend

A360557 Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

4, 10, 15, 22, 24, 25, 33, 34, 36, 40, 46, 51, 54, 55, 56, 62, 69, 77, 82, 85, 88, 93, 94, 100, 104, 115, 118, 119, 121, 123, 134, 135, 136, 141, 146, 152, 155, 161, 166, 177, 184, 187, 194, 196, 201, 205, 206, 217, 218, 219, 220, 221, 225, 232, 235, 240, 248

Offset: 1

Gus Wiseman, Feb 17 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is in the sequence.

For mean instead of median complement we have A340610, counted by A168659.
For mean instead of median we have A360668, counted by A200727.
Positions of odd terms in A360555.
The complement is A360556 (without 1), counted by A360688.
These partitions are counted by A360691.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551, complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A287352 lists 0-prepended first differences of prime indices.
A325347 counts partitions with integer median, complement A307683.
A355536 lists first differences of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A000975, A026424, A078175, A316413, A360009, A360558, A360669, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],!IntegerQ[Median[Differences[Prepend[prix[#],0]]]]&]

A359905 Numbers whose prime indices and prime signature both have integer mean.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 91, 94, 97, 100, 101, 103, 105, 107, 109, 110, 111, 113, 115, 118, 121

Offset: 1

Gus Wiseman, Jan 25 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.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
     2: {1}          19: {8}            39: {2,6}
     3: {2}          21: {2,4}          41: {13}
     4: {1,1}        22: {1,5}          43: {14}
     5: {3}          23: {9}            46: {1,9}
     7: {4}          25: {3,3}          47: {15}
     8: {1,1,1}      27: {2,2,2}        49: {4,4}
     9: {2,2}        29: {10}           53: {16}
    10: {1,3}        30: {1,2,3}        55: {3,5}
    11: {5}          31: {11}           57: {2,8}
    13: {6}          32: {1,1,1,1,1}    59: {17}
    16: {1,1,1,1}    34: {1,7}          61: {18}
    17: {7}          37: {12}           62: {1,11}

A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 lists prime signature, mean A088529/A088530.
A316413 lists numbers whose prime indices have integer mean.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A327473, A348551, A359904, A359908, A360005, A360068, A360069.

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

Intersection of A316413 and A067340.

A360007 Positions of first appearances in the sequence giving the median of the prime indices of n (A360005(n)/2).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 23, 26, 29, 31, 37, 38, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 74, 79, 83, 86, 89, 97, 101, 103, 106, 107, 109, 113, 122, 127, 131, 137, 139, 142, 149, 151, 157, 158, 163, 167, 173, 178, 179, 181, 191, 193, 197, 199, 202

Offset: 1

Gus Wiseman, Jan 24 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Positions of first appearances in A360005.
The unsorted version is A360006.
For mean instead of median we have A360008.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 counts partitions by median, cf. A359901, A359902.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A359889, A359890, A360009.

nn=1000;
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seq=Table[If[n==1,1,2*Median[prix[n]]],{n,nn}];
Select[Range[nn],FreeQ[seq[[Range[#-1]]],seq[[#]]]&]

Consists of 1, the primes, and all odd-indexed primes times 2.

A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 100, 112, 125, 180, 250, 252, 300, 352, 360, 392, 396, 405, 450, 468, 504, 540, 588, 600, 612, 675, 684, 720, 756, 792, 828, 832, 882, 900, 936, 1008, 1044, 1116, 1125, 1176, 1188, 1200, 1224, 1332, 1350, 1368, 1372, 1404, 1440, 1452, 1476

Offset: 1

Gus Wiseman, Feb 10 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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    9: {2,2}
   12: {1,1,2}
   18: {1,2,2}
   40: {1,1,1,3}
  100: {1,1,3,3}
  112: {1,1,1,1,4}
  125: {3,3,3}
  180: {1,1,2,2,3}
  250: {1,3,3,3}
  252: {1,1,2,2,4}
  300: {1,1,2,3,3}
  352: {1,1,1,1,1,5}
  360: {1,1,1,2,2,3}
For example, the prime indices of 756 are {1,1,2,2,2,4} with distinct parts {1,2,4} with median 2 and multiplicities {1,2,3} with median 2, so 756 is in the sequence.

Without taking median we have A109298, unordered A109297.
For mean instead of median we have A324570, counted by A114638.
For indices instead of multiplicities we have A360249, counted by A360245.
For indices instead of distinct indices we have A360454, counted by A360456.
These partitions are counted by A360455.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, length A001222, sum A056239.
A240219 counts partitions with mean equal to median, ranks A359889.
A316413 = numbers whose prime indices have integer mean, distinct A326621.
A325347 = partitions with integer median, strict A359907, ranks A359908.
A326567/A326568 gives mean of prime indices.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median.
A360005 gives median of prime indices (times two).
Cf. A000975, A359890, A359903, A360068, A360244, A360247, A360248.

Select[Range[100],#==1||Median[Last/@FactorInteger[#]]== Median[PrimePi/@First/@FactorInteger[#]]&]

A359913 Numbers whose multiset of prime factors has integer median.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jan 25 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The terms together with their prime factors begin:
   2: {2}
   3: {3}
   4: {2,2}
   5: {5}
   7: {7}
   8: {2,2,2}
   9: {3,3}
  11: {11}
  12: {2,2,3}
  13: {13}
  15: {3,5}
  16: {2,2,2,2}
  17: {17}
  18: {2,3,3}
  19: {19}
  20: {2,2,5}
  21: {3,7}
  23: {23}
  24: {2,2,2,3}

Prime factors are listed by A027746.
The complement is A072978, for prime indices A359912.
For mean instead of median we have A078175, for prime indices A316413.
For prime indices instead of factors we have A359908, counted by A325347.
Positions of even terms in A360005.
A067340 lists numbers whose prime signature has integer mean.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, strict A359907.
A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A067538, A175352, A348551, A359890, A359905, A360007, A360006, A360069.

Select[Range[2,100],IntegerQ[Median[Flatten[ConstantArray@@@FactorInteger[#]]]]&]

A360553 Numbers > 1 whose unordered prime signature has integer median.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A067340 in having 60.
A number's unordered prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The unordered prime signature of 60 is {1,1,2}, with median 1, so 60 is in the sequence.
The unordered prime signature of 1260 is {1,1,2,2}, with median 3/2, so 1260 is not in the sequence.

For mean instead of median we have A067340, complement A070011.
Positions of even terms in A360460.
The complement is A360554 (without 1).
These partitions are counted by A360687.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A112798 lists prime indices, length A001222, sum A056239.
A124010 lists prime signature.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360454 = numbers whose prime indices and signature have the same median.
Cf. A000975, A026424, A078174, A078175, A316413, A326621, A360453.

Select[Range[2,100],IntegerQ[Median[Last/@FactorInteger[#]]]&]

A360687 Number of integer partitions of n whose multiplicities have integer median.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676

Offset: 1

Gus Wiseman, Feb 20 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).

The case of an odd number of multiplicities is A090794.
For mean instead of median we have A360069, ranks A067340.
These partitions have ranks A360553.
The complement is counted by A360690, ranks A360554.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A360688 Number of integer partitions of n with integer median of 0-appended first differences.

Original entry on oeis.org

1, 1, 3, 4, 5, 7, 12, 18, 25, 32, 46, 62, 79, 109, 142, 189, 240, 322, 405, 522, 671, 853, 1053, 1345, 1653, 2081, 2551, 3174, 3878, 4826, 5851, 7219, 8747, 10712, 12936, 15719, 18876, 22872, 27365, 32926, 39253, 47070, 55857, 66676, 79029, 93864, 110832

Offset: 1

Gus Wiseman, Feb 20 2023

nonn

Includes all partitions of odd length (A027193).

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)        (8)
            (21)   (22)    (41)     (42)      (43)       (44)
            (111)  (211)   (221)    (222)     (61)       (62)
                   (1111)  (311)    (321)     (322)      (332)
                           (11111)  (411)     (331)      (422)
                                    (21111)   (421)      (431)
                                    (111111)  (511)      (521)
                                              (3211)     (611)
                                              (22111)    (2222)
                                              (31111)    (3221)
                                              (211111)   (4211)
                                              (1111111)  (22211)
                                                         (32111)
                                                         (41111)
                                                         (221111)
                                                         (311111)
                                                         (2111111)
                                                         (11111111)
For example, the partition y = (3,2,2,1) has 0-appended parts (3,2,2,1,0), with differences (1,0,1,1), and the multiset {0,1,1,1} has median 1, so y is counted under a(8).

The case of median 0 is A360254, ranks A360558.
These partitions have ranks A360556, complement A360557.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A237363, A240219, A360455, A360555.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Differences[Prepend[Reverse[#],0]]]]&]],{n,30}]

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]]&]

A360008 Positions of first appearances in the sequence giving the mean of prime indices (A326567/A326568).

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 12, 13, 14, 17, 18, 19, 23, 24, 26, 29, 31, 37, 38, 41, 42, 43, 47, 48, 52, 53, 54, 58, 59, 61, 67, 71, 72, 73, 74, 76, 79, 83, 86, 89, 92, 96, 97, 101, 103, 104, 106, 107, 108, 109, 113, 122, 124, 127, 131, 137, 139, 142, 148, 149, 151, 152

Offset: 1

Gus Wiseman, Jan 24 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:
    1: {}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   24: {1,1,1,2}

Positions of first appearances in A326567/A326568.
The version for median instead of mean is A360007, unsorted A360006.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 lists prime indices, length A001222, sum A056239.
A316413 lists numbers whose prime indices have integer mean.
A326567/A326568 gives mean of prime indices.
A359908 = numbers w/ integer median of prime indices, complement A359912.
Cf. A026424, A051293, A327473, A348551, A359889, A360005.

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

cp's OEIS Frontend

A360557 Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

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A359905 Numbers whose prime indices and prime signature both have integer mean.

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A360007 Positions of first appearances in the sequence giving the median of the prime indices of n (A360005(n)/2).

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A360453 Numbers for which the prime multiplicities (or sorted signature) have the same median as the distinct prime indices.

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A359913 Numbers whose multiset of prime factors has integer median.

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A360553 Numbers > 1 whose unordered prime signature has integer median.

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A360687 Number of integer partitions of n whose multiplicities have integer median.

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A360688 Number of integer partitions of n with integer median of 0-appended first differences.

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

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