A360254 -id:A360254 - cp's OEIS frontend

A111907 Numbers k such that the same number of primes, among primes <= the largest prime dividing k, divide k as do not.

1, 3, 9, 14, 21, 27, 28, 35, 56, 63, 78, 81, 98, 112, 130, 147, 156, 175, 182, 189, 195, 196, 224, 234, 243, 245, 260, 273, 286, 312, 364, 392, 429, 441, 448, 455, 468, 520, 567, 570, 572, 585, 624, 650, 686, 702, 715, 728, 729, 784, 798, 819, 875, 896, 936

Offset: 1

Leroy Quet, Aug 19 2005

nonn

Also numbers whose greatest prime index (A061395) is twice their number of distinct prime factors (A001221). - Gus Wiseman, Mar 19 2023

			28 is included because 7 is the largest prime dividing 28. And of the primes <= 7 (2,3,5,7), 2 and 7 (2 primes) divide 28 and 3 and 5 (also 2 primes) do not divide 28.
From _Gus Wiseman_, Mar 19 2023: (Start)
The terms together with their prime indices begin:
    1: {}
    3: {2}
    9: {2,2}
   14: {1,4}
   21: {2,4}
   27: {2,2,2}
   28: {1,1,4}
   35: {3,4}
   56: {1,1,1,4}
   63: {2,2,4}
   78: {1,2,6}
   81: {2,2,2,2}
   98: {1,4,4}
  112: {1,1,1,1,4}
  130: {1,3,6}
  147: {2,4,4}
  156: {1,1,2,6}
For example, 156 is included because it has prime indices {1,1,2,6}, with distinct parts {1,2,6} and distinct non-parts {3,4,5}, both of length 3. Alternatively, 156 has greatest prime index 6 and omega 3, and 6 = 2*3.
(End)

John Tyler Rascoe, Table of n, a(n) for n = 1..1000

For length instead of maximum we have A067801.
These partitions are counted by A239959.
A001222 (bigomega) counts prime factors, distinct A001221 (omega).
A061395 gives greatest prime index.
A112798 lists prime indices, sum A056239.
Comparing twice the number of distinct parts to greatest part:
              less: A360254, ranks A111906
             equal: A239959, ranks A111907
           greater: A237365, ranks A111905
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A046660, A067340, A324517, A324521, A324562, A361205, A361393.

Select[Range[100],2*PrimeNu[#]==PrimePi[FactorInteger[#][[-1,1]]]&] (* Gus Wiseman, Mar 19 2023 *)

{m=950;v=vector(m);for(n=1,m,f=factor(n)[,1]~;c=0;pc=0;forprime(p=2,vecmax(f), j=1;s=length(f);while(j<=s&&p!=f[j],j++);if(j<=s,c++);pc++);v[n]=sign(pc-2*c)); for(n=1,m,if(v[n]==0,print1(n,",")))} \\ Klaus Brockhaus, Aug 21 2005

from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    yield 1
    for k in count(3):
        f = [sieve.search(i)[0] for i in factorint(k)]
        if 2*len(f) == f[-1]:
            yield k
A111907_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 20 2024

More terms from Klaus Brockhaus, Aug 21 2005

A360672 Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 0, 0, 3, 6, 4, 1, 0, 1, 0, 0, 1, 7, 7, 5, 1, 0, 1, 0, 0, 1, 4, 8, 10, 5, 1, 0, 1, 0, 0, 0, 3, 6, 14, 11, 6, 1, 0, 1, 0, 0, 0, 1, 5, 12, 16, 14, 6, 1, 0

Offset: 0

Gus Wiseman, Feb 27 2023

Also the number of integer partitions of n whose right half (inclusive) sums to n-k.

			Triangle begins:
  1
  1  0
  1  1  0
  1  1  1  0
  1  0  3  1  0
  1  0  2  3  1  0
  1  0  1  4  4  1  0
  1  0  0  3  6  4  1  0
  1  0  0  1  7  7  5  1  0
  1  0  0  1  4  8 10  5  1  0
  1  0  0  0  3  6 14 11  6  1  0
  1  0  0  0  1  5 12 16 14  6  1  0
  1  0  0  0  1  2 12 14 23 16  7  1  0
  1  0  0  0  0  2  7 13 24 27 19  7  1  0
  1  0  0  0  0  1  5  9 24 30 35 21  8  1  0
  1  0  0  0  0  1  3  7 17 31 42 40 25  8  1  0
  1  0  0  0  0  0  2  4 16 23 46 51 51 27  9  1  0
  1  0  0  0  0  0  1  3 10 21 37 57 69 57 31  9  1  0
  1  0  0  0  0  0  1  2  7 15 34 47 83 81 69 34 10  1  0
For example, row n = 9 counts the following partitions:
  (9)  .  .  (333)  (432)        (54)        (63)      (72)    (81)
                    (441)        (522)       (621)     (711)
                    (22221)      (531)       (3321)    (4311)
                    (111111111)  (3222)      (4221)    (5211)
                                 (32211)     (33111)   (6111)
                                 (2211111)   (42111)
                                 (3111111)   (51111)
                                 (21111111)  (222111)
                                             (321111)
                                             (411111)
For example, the partition y = (3,2,2,1,1) has left half (exclusive) (3,2), with sum 5, so y is counted under T(9,5).

Row sums are A000041.
Column sums are A360673, inclusive A360671.
The central diagonal T(2n,n) is A360674, ranks A360953.
The left inclusive version is A360675 with rows reversed.
A008284 counts partitions by length.
A359893 and A359901 count partitions by median.
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. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616, A360682, A360686.

Table[Length[Select[IntegerPartitions[n], Total[Take[#,Floor[Length[#]/2]]]==k&]],{n,0,10},{k,0,n}]

A360675 Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 3, 3, 0, 0, 0, 1, 3, 5, 2, 0, 0, 0, 1, 4, 6, 4, 0, 0, 0, 0, 1, 4, 9, 5, 3, 0, 0, 0, 0, 1, 5, 10, 10, 4, 0, 0, 0, 0, 0, 1, 5, 13, 12, 9, 2, 0, 0, 0, 0, 0, 1, 6, 15, 18, 11, 5, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Feb 27 2023

Also the number of integer partitions of n whose left half (inclusive) sums to n-k.

			Triangle begins:
  1
  1  0
  1  1  0
  1  2  0  0
  1  2  2  0  0
  1  3  3  0  0  0
  1  3  5  2  0  0  0
  1  4  6  4  0  0  0  0
  1  4  9  5  3  0  0  0  0
  1  5 10 10  4  0  0  0  0  0
  1  5 13 12  9  2  0  0  0  0  0
  1  6 15 18 11  5  0  0  0  0  0  0
  1  6 18 22 20  6  4  0  0  0  0  0  0
  1  7 20 29 26 13  5  0  0  0  0  0  0  0
  1  7 24 34 37 19 11  2  0  0  0  0  0  0  0
  1  8 26 44 46 30 16  5  0  0  0  0  0  0  0  0
  1  8 30 50 63 40 27  8  4  0  0  0  0  0  0  0  0
  1  9 33 61 75 61 36 15  6  0  0  0  0  0  0  0  0  0
  1  9 37 70 96 75 61 21 12  3  0  0  0  0  0  0  0  0  0
For example, row n = 9 counts the following partitions:
  (9)  (81)   (72)     (63)       (54)
       (441)  (432)    (333)      (3222)
       (531)  (522)    (3321)     (21111111)
       (621)  (4311)   (4221)     (111111111)
       (711)  (5211)   (22221)
              (6111)   (222111)
              (32211)  (321111)
              (33111)  (411111)
              (42111)  (2211111)
              (51111)  (3111111)
For example, the partition y = (3,2,2,1,1) has right half (exclusive) (1,1), with sum 2, so y is counted under T(9,2).

The central diagonal T(2n,n) is A000005.
Row sums are A000041.
Diagonal sums are A360671, exclusive A360673.
The right inclusive version is A360672 with rows reversed.
The left version has central diagonal A360674, ranks A360953.
A008284 counts partitions by length.
A359893 and A359901 count partitions by median.
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. A027193, A237363, A307683, A325347, A360005, A360071, A360254, A360616.

Table[Length[Select[IntegerPartitions[n], Total[Take[#,-Floor[Length[#]/2]]]==k&]],{n,0,18},{k,0,n}]

A360555 Two times the median of the first differences of the 0-prepended prime indices of n > 1.

Original entry on oeis.org

2, 4, 1, 6, 2, 8, 0, 2, 3, 10, 2, 12, 4, 3, 0, 14, 2, 16, 2, 4, 5, 18, 1, 3, 6, 0, 2, 20, 2, 22, 0, 5, 7, 4, 1, 24, 8, 6, 1, 26, 2, 28, 2, 2, 9, 30, 0, 4, 2, 7, 2, 32, 1, 5, 1, 8, 10, 34, 2, 36, 11, 4, 0, 6, 2, 38, 2, 9, 2, 40, 0, 42, 12, 2, 2, 5, 2, 44, 0, 0

Offset: 2

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.

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 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so a(1617) = 3.

The version for divisors is A063655.
Differences of 0-prepended prime indices are listed by A287352.
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 factors is A360459.
The version for prime multiplicities is A360460.
Positions of even terms are A360556
Positions of odd terms are A360557
Positions of 0's are A360558, counted by A360254.
For mean instead of two times median we have A360614/A360615.
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, A359907, A359908, A359912, A360550.

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

A237363 Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 6, 6, 10, 13, 20, 26, 39, 50, 71, 87, 121, 156, 208, 265, 348, 440, 566, 712, 906, 1131, 1424, 1766, 2224, 2738, 3390, 4168, 5130, 6266, 7664, 9312, 11332, 13723, 16603, 20004, 24112, 28942, 34708, 41522, 49612, 59031, 70308, 83479, 98992

Offset: 0

Clark Kimberling, Feb 06 2014

nonn

a(n) + A237365(n) = A000041(n).

Also the number of integer partitions of n whose median difference is 0. For example, the partition (2,2,2,1,1) is counted because its multiset of differences {0,0,0,1} has median 0. - Gus Wiseman, Mar 18 2023

			Among the 22 partitions of 8, these qualify:  [5,1,1,1], [4,4], [4,1,1,1,1], [3,3,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1], and the remaining 12 do not, so that a(8) = 10.

Alois P. Heinz, Table of n, a(n) for n = 0..800

These partitions have ranks A361204.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts, reverse A058398.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
Comparing twice the number of distinct parts to the number of parts:
              less: A360254, ranks A360558
             equal: A239959, ranks A067801
           greater: A237365, ranks A361393
     less or equal: A237363, ranks A361204
  greater or equal: A361394, ranks A361395
Cf. A000975, A027193, A114638, A240219, A325347, A360005, A360244.

z = 50; t = Map[Length[Select[IntegerPartitions[#], 2*Length[DeleteDuplicates[#]] <= Length[#] &]] &, Range[z]] (*A237363*)
Table[PartitionsP[n] - t[[n]], {n, 1, z}] (*A237365*) (* Peter J. C. Moses, Feb 06 2014 *)
Table[Length[Select[IntegerPartitions[n],Median[Differences[#]]==0&]],{n,0,30}] (* Gus Wiseman, Mar 18 2023 *)

A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 11, 15, 20, 30, 38, 49, 65, 83, 108, 139, 178, 224, 286, 358, 437, 550, 684, 837, 1037, 1269, 1553, 1889, 2295, 2770, 3359, 4035, 4843, 5808, 6951, 8312, 9902, 11752, 13958, 16531, 19541, 23037, 27162, 31911, 37488, 43950, 51463, 60127, 70229

Offset: 0

Gus Wiseman, Mar 17 2023

nonn

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (22)   (32)    (33)    (43)
                   (31)   (41)    (42)    (52)
                   (211)  (221)   (51)    (61)
                          (311)   (321)   (322)
                          (2111)  (411)   (331)
                                  (2211)  (421)
                                  (3111)  (511)
                                          (2221)
                                          (3211)
                                          (4111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The complement is counted by A360254, ranks A360558.
These partitions have ranks A361395.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, reverse A058398.
A067538 counts partitions with integer mean, strict A102627.
A116608 counts partitions by number of distinct parts.
Cf. A106529, A144300, A237363, A237832, A240219, A316413, A349156.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>=0, 1, 0),
     `if`(i<1, 0, add(b(n-i*j, i-1, t+`if`(j>0, 2, 0)-j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2023

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

A360558 Numbers whose multiset of prime factors (or indices, see A112798) has more adjacent equalities (or parts that have appeared before) than distinct parts.

Original entry on oeis.org

8, 16, 27, 32, 48, 64, 72, 80, 81, 96, 108, 112, 125, 128, 144, 160, 162, 176, 192, 200, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 343, 352, 368, 384, 392, 400, 405, 416, 432, 448, 464, 480, 486, 496, 500, 512, 544, 567, 576, 592, 608, 625, 640, 648

Offset: 1

Gus Wiseman, Feb 20 2023

nonn

No terms are squarefree.

Also numbers whose first differences of 0-prepended prime indices have median 0.

			The terms together with their prime indices begin:
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    32: {1,1,1,1,1}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    81: {2,2,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   112: {1,1,1,1,4}
   125: {3,3,3}
For example, the prime indices of 720 are {1,1,1,1,2,2,3} with 4 adjacent equalities and 3 distinct parts, so 720 is in the sequence.

For equality we have A067801.
These partitions are counted by A360254.
A112798 lists prime indices, length A001222, sum A056239.
A326567/A326568 gives mean of prime indices.
A360005 gives median of prime indices (times 2).
Cf. A000975, A027193, A067340, A237363, A317090, A360248, A360249, A360454, A360555, A360556.

Select[Range[100],PrimeOmega[#]>2*PrimeNu[#]&]

A001222(a(n)) > 2*A001221(a(n)).

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}]

A361205 a(n) = 2*omega(n) - bigomega(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, -1, 0, 2, 1, 1, 1, 2, 2, -2, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, -1, 1, 1, 3, 1, -3, 2, 2, 2, 0, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, -1, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 2, 1, 2, 1, -4, 2, 3, 1, 1, 2, 3, 1, -1, 1, 2, 1, 1, 2, 3, 1, -1, -2, 2, 1, 2

Offset: 1

Gus Wiseman, Mar 16 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Without doubling omega we have -A046660.
Positions of 0's are A067801, counted by A239959.
Positions of negative terms are A360558, counted by A360254.
Positions of nonpositive terms are A361204, counted by A237363.
Positions of positive terms are A361393, counted by A237365.
Positions of nonnegative terms are A361395, counted by A361394.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
Cf. A061395, A067340, A111907, A324517, A324522, A326567/A326568.
Cf. A077761, A083342, A136141.

Table[2*PrimeNu[n]-PrimeOmega[n],{n,100}]

Additive with a(p^e) = 2 - e. - Amiram Eldar, Mar 26 2023

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = 2*A077761 - A083342 = A077761 - A136141 = -0.511659... . - Amiram Eldar, Oct 01 2023

A360674 Number of integer partitions of 2n whose left half (exclusive) and right half (inclusive) both sum to n.

Original entry on oeis.org

1, 1, 3, 4, 7, 6, 12, 9, 16, 15, 21, 16, 34, 22, 33, 36, 47, 36, 62, 44, 75, 68, 78, 68, 120, 93, 113, 117, 151, 122, 195, 148, 209, 197, 220, 226, 315, 249, 304, 309, 402, 332, 463, 387, 496, 515, 539, 514, 712, 609, 738, 723, 845, 774, 983, 914, 1111

Offset: 0

Gus Wiseman, Mar 04 2023

nonn

Of course, only one of the two conditions is necessary.

			The a(1) = 1 through a(6) = 12 partitions:
  (11)  (22)    (33)      (44)        (55)          (66)
        (211)   (321)     (422)       (532)         (633)
        (1111)  (21111)   (431)       (541)         (642)
                (111111)  (2222)      (32221)       (651)
                          (22211)     (211111111)   (3333)
                          (2111111)   (1111111111)  (33222)
                          (11111111)                (33321)
                                                    (42222)
                                                    (222222)
                                                    (2222211)
                                                    (21111111111)
                                                    (111111111111)
For example, the partition y = (3,2,2,2,1) has halves (3,2) and (2,2,1), both with sum 5, so y is counted under a(5).

The even-length case is A000005.
Central diagonal of A360672.
These partitions have ranks A360953.
A008284 counts partitions by length, row sums A000041.
A359893 and A359901 count partitions by median.
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. A237363, A360254, A360671, A360673, A360682.

Table[Length[Select[IntegerPartitions[2n], Total[Take[#,Floor[Length[#]/2]]]==n&]],{n,0,15}]

def accel_asc(n):
    a = [0 for i in range(n + 1)]
    k = 1
    y = n - 1
    while k != 0:
        x = a[k - 1] + 1
        k -= 1
        while 2 * x <= y:
            a[k] = x
            y -= x
            k += 1
        l = k + 1
        while x <= y:
            a[k] = x
            a[l] = y
            yield a[:k + 2]
            x += 1
            y -= 1
        a[k] = x + y
        y = x + y - 1
        yield a[:k + 1]
for y in range(1000):
    num = 0
    for x in accel_asc(2*y):
        stop = len(x)//2+1
        if len(x) % 2 == 0:
            stop -= 1
        right = x[0:stop]
        left = x[stop:]
        if sum(right) == sum(left):
            num += 1
    print(y,num)
# David Consiglio, Jr., Mar 09 2023

a(n) = A360672(2n,n).

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

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