A360688 -id:A360688 - cp's OEIS frontend

A360556 Numbers > 1 whose first differences of 0-prepended prime indices have integer median.

2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 26, 27, 28, 29, 30, 31, 32, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84, 86, 87, 89

Offset: 1

Gus Wiseman, Feb 16 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 not in the sequence.

For mean instead of median we have A340610.
Positions of even terms in A360555.
The complement is A360557 (without 1).
These partitions are counted by A360688.
- 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.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
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]]]]&]

A360254 Number of integer partitions of n with more adjacent equal parts than distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 7, 10, 12, 18, 28, 36, 52, 68, 92, 119, 161, 204, 269, 355, 452, 571, 738, 921, 1167, 1457, 1829, 2270, 2834, 3483, 4314, 5300, 6502, 7932, 9665, 11735, 14263, 17227, 20807, 25042, 30137, 36099, 43264, 51646, 61608, 73291, 87146, 103296

Offset: 0

Gus Wiseman, Feb 20 2023

nonn

None of these partitions is strict.

Also the number of integer partitions of n which, after appending 0, have first differences of median 0.

			The a(3) = 1 through a(9) = 10 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (21111)   (31111)    (22211)     (22221)
                          (111111)  (211111)   (41111)     (33111)
                                    (1111111)  (221111)    (51111)
                                               (311111)    (222111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)
For example, the partition y = (4,4,3,1,1,1,1) has 0-appended differences (0,1,2,0,0,0,0), with median 0, so y is counted under a(15).

The non-prepended version is A237363.
These partitions have ranks A360558.
For any integer median (not just 0) we have A360688.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A067538, A102627, A240219, A359894, A360071, A360244, A360555, A360556.

Table[Length[Select[IntegerPartitions[n], Length[#]>2*Length[Union[#]]&]],{n,0,30}]

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]]]]&]

A360614 Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 1, 1, 3, 5, 2, 6, 2, 3, 1, 7, 2, 8, 1, 2, 5, 9, 1, 3, 3, 2, 4, 10, 1, 11, 1, 5, 7, 2, 1, 12, 4, 3, 3, 13, 4, 14, 5, 1, 9, 15, 2, 2, 1, 7, 2, 16, 1, 5, 1, 4, 5, 17, 3, 18, 11, 4, 1, 3, 5, 19, 7, 9, 4, 20, 2, 21, 6, 1, 8, 5, 2, 22, 3, 1, 13, 23, 1, 7, 7, 5, 5, 24, 3, 3, 3, 11, 15, 4, 1, 25, 4, 5, 3

Offset: 1

Gus Wiseman, Feb 19 2023

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 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 3.

Positions of 1's are A340609, a superset of A106529.
For twice median instead of mean we have A360555.
The denominator is A360615.
A112798 lists prime indices, length A001222, sum A056239, max A061395.
A124010 gives prime signature, mean A088529/A088530.
A316413 lists numbers with integer mean prime index, complement A348551.
A326567/A326568 gives mean of prime indices.
Cf. A334201, A340610, A360008, A360556, A360557, A360688, A366785.

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

A360614(n) = if(1==n,0, my(u=primepi(vecmax(factor(n)[, 1]))); (u/gcd(u, bigomega(n)))); \\ Antti Karttunen, Oct 23 2023

Numerator of A061395(n)/A001222(n).

a(1) = 0; and for n >= 1, a(n) = A061395(n) / A366785(n) = A061395(n) / gcd(A001222(n), A061395(n)). - Antti Karttunen, Oct 23 2023

Data section extended up to a(100) by Antti Karttunen, Oct 23 2023

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}]

A360686 Number of integer partitions of n whose distinct parts have integer median.

Original entry on oeis.org

1, 2, 2, 4, 3, 8, 7, 16, 17, 31, 35, 60, 67, 99, 121, 170, 200, 270, 328, 436, 522, 674, 828, 1061, 1292, 1626, 1983, 2507, 3035, 3772, 4582, 5661, 6801, 8358, 10059, 12231, 14627, 17702, 21069, 25423, 30147, 36100, 42725, 50936, 60081, 71388, 84007, 99408

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)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (421)      (53)
                    (1111)           (51)      (511)      (62)
                                     (222)     (3211)     (71)
                                     (321)     (31111)    (422)
                                     (3111)    (1111111)  (431)
                                     (111111)             (521)
                                                          (2222)
                                                          (3221)
                                                          (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).

For all parts: A325347, strict A359907, ranks A359908, complement A307683.
For mean instead of median: A360241, ranks A326621.
These partitions have ranks A360550, complement A360551.
For multiplicities instead of distinct parts: A360687.
The complement is counted by A360689.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A027193 counts odd-length partitions, strict A067659, ranks A026424.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A240219, A359906, A360005, A360071, A360244, A360245, A360556, A360688.

Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Union[#]]]&]],{n,30}]

A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 42, 49, 60, 66, 70, 78, 84, 90, 102, 105, 114, 120, 126, 132, 138, 140, 150, 154, 156, 168, 174, 186, 198, 204, 210, 222, 228, 234, 246, 258, 264, 270, 276, 280, 282, 286, 294, 306, 308, 312, 315, 318, 330, 342, 348, 350, 354, 366, 372, 378, 385

Offset: 1

Gus Wiseman, Feb 19 2023

nonn

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 terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
   30: {1,2,3}
   42: {1,2,4}
   49: {4,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with median 1. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with median 1/2. So 2760 is not in the sequence.

For distinct prime indices instead of 0-prepended differences: A360453.
For mean instead of median we have A360680.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
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.
Multisets with integer median:
- 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.
Cf. A000975, A026424, A316413, A340610, A359904, A360006, A360248, A360558, A360687, A360688.

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

A360682 Number of integer partitions of n of length > 2 whose second differences have median 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 5, 4, 10, 13, 18, 23, 44, 44, 72, 98, 132, 162, 241, 277, 394, 497, 643, 800, 1076, 1287, 1660, 2078, 2604, 3192, 4065, 4892, 6113, 7490, 9166, 11110, 13717, 16429, 20033, 24201, 29143, 34945, 42251, 50219, 60253, 71852, 85503, 101501, 120899

Offset: 0

Gus Wiseman, Feb 19 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(3) = 1 through a(9) = 13 partitions:
  (111)  (1111)  (11111)  (222)     (22111)    (2222)      (333)
                          (321)     (31111)    (3221)      (432)
                          (2211)    (211111)   (3311)      (531)
                          (21111)   (1111111)  (22211)     (22221)
                          (111111)             (32111)     (33111)
                                               (41111)     (51111)
                                               (221111)    (222111)
                                               (311111)    (321111)
                                               (2111111)   (411111)
                                               (11111111)  (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)

For first differences we have A237363.
For sum instead of median we have A360683.
For mean instead of median we have A360683 - A008619.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives median of prime indices (times two).
Cf. A000975, A027193, A067538, A114638, A307683, A359908, A360245, A360254, A360687, A360688.

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

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465

Offset: 1

Gus Wiseman, Feb 22 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) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
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 not counted under a(8).

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
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.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

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

A360691 Number of integer partitions of n with non-integer median of 0-prepended first differences.

Original entry on oeis.org

0, 1, 0, 1, 2, 4, 3, 4, 5, 10, 10, 15, 22, 26, 34, 42, 57, 63, 85, 105, 121, 149, 202, 230, 305, 355, 459, 544, 687, 778, 991, 1130, 1396, 1598, 1947, 2258, 2761, 3143, 3820, 4412, 5330, 6104, 7404, 8499, 10105, 11694, 13922, 15917, 18904, 21646, 25462, 29213

Offset: 1

Gus Wiseman, Feb 22 2023

nonn

All of these partitions have even length.

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) = 0 through a(10) = 10 partitions:
  .  (11)  .  (31)  (32)    (33)    (52)    (53)    (54)      (55)
                    (2111)  (51)    (2221)  (71)    (72)      (73)
                            (2211)  (4111)  (3311)  (3222)    (91)
                            (3111)          (5111)  (6111)    (3322)
                                                    (321111)  (3331)
                                                              (4411)
                                                              (5311)
                                                              (7111)
                                                              (322111)
                                                              (421111)

For median 0 we have A360254, ranks A360558.
These partitions have ranks A360557, complement A360556.
The complement is counted by A360688.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A027193, A237363, A240219, A359907, A359908, A360455, A360555.

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

cp's OEIS Frontend

A360556 Numbers > 1 whose first differences of 0-prepended prime indices have integer median.

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A360254 Number of integer partitions of n with more adjacent equal parts than distinct parts.

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

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A360614 Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

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

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A360686 Number of integer partitions of n whose distinct parts have integer median.

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A360681 Numbers for which the prime signature has the same median as the first differences of 0-prepended prime indices.

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A360682 Number of integer partitions of n of length > 2 whose second differences have median 0.

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A360690 Number of integer partitions of n with non-integer median of multiplicities.

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