A360457 -id:A360457 - cp's OEIS frontend

A360677 Sum of the right half (exclusive) of the prime indices of n.

0, 0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 4, 3, 2, 0, 2, 0, 3, 4, 5, 0, 3, 3, 6, 2, 4, 0, 3, 0, 2, 5, 7, 4, 4, 0, 8, 6, 4, 0, 4, 0, 5, 3, 9, 0, 3, 4, 3, 7, 6, 0, 4, 5, 5, 8, 10, 0, 5, 0, 11, 4, 3, 6, 5, 0, 7, 9, 4, 0, 4, 0, 12, 3, 8, 5, 6, 0, 4, 4, 13, 0, 6, 7

Offset: 1

Gus Wiseman, Mar 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 prime indices of 810 are {1,2,2,2,2,3}, with right half (exclusive) {2,2,3}, so a(810) = 7.
The prime indices of 3675 are {2,3,3,4,4}, with right half (exclusive) {4,4}, so a(3675) = 8.

Positions of 0's are 1 and A000040.
Positions of last appearances are A004171.
Positions of first appearances are A100484.
These partitions are counted by A360672.
The value k > 0 appears A360673(k) times, inclusive A360671.
The left version is A360676.
The inclusive version is A360679.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A001248, A026424, A359908, A359912, A360006, A360457.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],-Floor[Length[prix[n]]/2]]],{n,100}]

Last position of k is 2^(2k+1).
A360676(n) + A360679(n) = A001222(n).
A360677(n) + A360678(n) = A001222(n).

A360678 Sum of the left half (inclusive) of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 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 prime indices of 810 are {1,2,2,2,2,3}, with left half (inclusive) {1,2,2}, so a(810) = 5.
The prime indices of 3675 are {2,3,3,4,4}, with left half (inclusive) {2,3,3}, so a(3675) = 8.

Positions of first appearances are 1 and A001248.
Positions of 1's are A001747.
These partitions are counted by A360675 with rows reversed.
The exclusive version is A360676.
The right version is A360679.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A026424, A280076, A359912, A360006, A360457.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],Ceiling[Length[prix[n]]/2]]],{n,100}]

A360676(n) + A360679(n) = A001222(n).

A360677(n) + A360678(n) = A001222(n).

A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 4, 26, 9, 8, 4, 34, 6, 38, 4, 10, 13, 46, 4, 10, 15, 6, 4, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 4, 82, 6, 86, 4, 6, 25, 94, 4, 14, 10, 20, 4, 106, 6, 16, 4, 22, 31, 118, 5, 122, 33, 6, 4, 18, 6, 134, 4, 26, 10, 142, 4, 146

Offset: 1

Gus Wiseman, Feb 14 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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

			The prime factors of 60 are {2,2,3,5}, with median 5/2, so a(60) = 5.

The union is 2 followed by A014091, complement of A014092.
The prime factors themselves are listed by A027746, distinct A027748.
The version for divisors is A063655.
Positions of odd terms are A072978 (except 1).
For mean instead of twice median: A123528/A123529, distinct A323171/A323172.
Positions of even terms are A359913 (and 1).
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for distinct prime factors is A360458.
The version for prime multiplicities is A360460.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A027336, A078174, A316413, A359907, A359908, A360006, A360007, A360248, A360552.

Table[2*Median[Join@@ConstantArray@@@FactorInteger[n]],{n,100}]

A360458 Two times the median of the set of distinct prime factors of n; a(1) = 2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 5, 26, 9, 8, 4, 34, 5, 38, 7, 10, 13, 46, 5, 10, 15, 6, 9, 58, 6, 62, 4, 14, 19, 12, 5, 74, 21, 16, 7, 82, 6, 86, 13, 8, 25, 94, 5, 14, 7, 20, 15, 106, 5, 16, 9, 22, 31, 118, 6, 122, 33, 10, 4, 18, 6, 134, 19, 26, 10, 142, 5

Offset: 1

Gus Wiseman, Feb 14 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). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

			The prime factors of 336 are {2,2,2,2,3,7}, with distinct parts {2,3,7}, with median 3, so a(336) = 6.

The union is 2 followed by A014091, complement of A014092.
Distinct prime factors are listed by A027748.
The version for divisors is A063655.
Positions of odd terms are A100367.
For mean instead of two times median we have A323171/A323172.
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for prime factors is A360459.
The version for prime multiplicities is A360460.
Positions of even terms are A360552.
The version for 0-prepended differences is A360555.
A112798 lists prime indices, length A001222, sum A056239.
A304038 lists distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A078174, A316413, A325347, A359907, A360006, A360248, A360453, A360550, A360551.

Table[2*Median[First/@FactorInteger[n]],{n,100}]

A360551 Numbers > 1 whose distinct prime indices have non-integer median.

Original entry on oeis.org

6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A325700 in having 330 and lacking 462.
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. Distinct prime indices are listed by A304038.
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 prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.

For mean instead of median we have the complement of A326621.
Positions of odd terms in A360457.
The complement (without 1) is A360550, counted by A360686.
- 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.
A304038 lists distinct prime indices, length A001221, sum A066328.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A316413, A348551, A360006, A360009, A360248, A360453.

Select[Range[2,100],!IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]

A360554 Numbers > 1 whose unordered prime signature has non-integer median.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 48, 50, 52, 63, 68, 72, 75, 76, 80, 92, 98, 99, 108, 112, 116, 117, 124, 147, 148, 153, 162, 164, 171, 172, 175, 176, 188, 192, 200, 207, 208, 212, 236, 242, 244, 245, 261, 268, 272, 275, 279, 284, 288, 292, 304, 316, 320, 325, 332, 333

Offset: 1

Gus Wiseman, Feb 16 2023

nonn

First differs from A187039 in having 2520 and lacking 1 and 12600.
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 2520 is {3,2,1,1}, with median 3/2, so 2520 is in the sequence.
The unordered prime signature of 12600 is {3,2,2,1}, with median 2, so 12600 is not in the sequence.

A subset of A030231.
For mean instead of median we have A070011.
Positions of odd terms in A360460.
The complement is A360553 (without 1), counted by A360687.
- 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.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A326619/A326620 gives mean of distinct prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A304038, A316413, A326621, A348551, A360006, A360009, A360248, A360453.

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

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

Original entry on oeis.org

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]]]]&]

A361858 Number of integer partitions of n such that the maximum is less than twice the median.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 8, 12, 15, 19, 22, 31, 34, 45, 55, 67, 78, 100, 115, 144, 170, 203, 238, 291, 337, 403, 473, 560, 650, 772, 889, 1046, 1213, 1414, 1635, 1906, 2186, 2533, 2913, 3361, 3847, 4433, 5060, 5808, 6628, 7572, 8615, 9835, 11158, 12698, 14394

Offset: 1

Gus Wiseman, Apr 02 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) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (322)      (71)
                                     (321)     (331)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
The partition y = (3,2,2,1) has maximum 3 and median 2, and 3 < 2*2, so y is counted under a(8).

For minimum instead of median we have A053263.
For length instead of median we have A237754.
Allowing equality gives A361848, strict A361850.
The equal version is A361849, ranks A361856.
For mean instead of median we have A361852.
Reversing the inequality gives A361857, ranks A361867.
The complement is counted by A361859, ranks A361868.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A027193, A237751, A237755, A237820, A237824, A240219, A361394, A361851, A361860, A361907.

Table[Length[Select[IntegerPartitions[n],Max@@#<2*Median[#]&]],{n,30}]

A361860 Number of integer partitions of n whose median part is the smallest.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 12, 15, 21, 25, 36, 44, 58, 72, 95, 117, 150, 185, 235, 289, 362, 441, 550, 670, 824, 1000, 1223, 1476, 1795, 2159, 2609, 3126, 3758, 4485, 5369, 6388, 7609, 9021, 10709, 12654, 14966, 17632, 20782, 24414, 28684, 33601, 39364, 45996

Offset: 1

Gus Wiseman, Apr 02 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) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For mean instead of median we have A000005.
For length instead of median we have A006141.
For maximum instead of median we have A053263.
For half-median we have A361861.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A027193, A053263, A067659, A111907, A116608, A118096, A237753, A240219, A359907, A361848, A361849.

Table[Length[Select[IntegerPartitions[n],Min@@#==Median[#]&]],{n,30}]

A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 10, 15, 23, 34, 46, 67, 90, 121, 164, 219, 285, 375, 483, 622, 799, 1017, 1284, 1621, 2033, 2537, 3158, 3915, 4832, 5953, 7303, 8930, 10896, 13248, 16071, 19451, 23482, 28272, 33977, 40736, 48741, 58201, 69367, 82506, 97986, 116139

Offset: 1

Gus Wiseman, Apr 02 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(4) = 1 through a(9) = 15 partitions:
  (211)  (311)   (411)    (421)     (422)      (522)
         (2111)  (3111)   (511)     (521)      (621)
                 (21111)  (3211)    (611)      (711)
                          (4111)    (4211)     (4221)
                          (22111)   (5111)     (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (33111)
                                    (311111)   (42111)
                                    (2111111)  (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 >= 2*2, so y is counted under a(10).

For length instead of median we have A237752.
For minimum instead of median we have A237821.
Reversing the inequality gives A361848.
The equal case is A361849, ranks A361856.
The unequal case is A361857, ranks A361867.
The complement is counted by A361858.
These partitions have ranks A361868.
For mean instead of median we have A361906.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A027193, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.

Table[Length[Select[IntegerPartitions[n],Max@@#>=2*Median[#]&]],{n,30}]

cp's OEIS Frontend

A360677 Sum of the right half (exclusive) of the prime indices of n.

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A360678 Sum of the left half (inclusive) of the prime indices of n.

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A360459 Two times the median of the multiset of prime factors of n; a(1) = 2.

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A360458 Two times the median of the set of distinct prime factors of n; a(1) = 2.

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A360551 Numbers > 1 whose distinct prime indices have non-integer median.

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A360554 Numbers > 1 whose unordered prime signature has non-integer median.

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

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A361858 Number of integer partitions of n such that the maximum is less than twice the median.

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A361860 Number of integer partitions of n whose median part is the smallest.

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