A102627 -id:A102627 - cp's OEIS frontend

A327782 Numbers n that cannot be written as a sum of two or more distinct parts with the largest part dividing n.

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211, 221

Offset: 1

Gus Wiseman, Sep 25 2019

nonn

After initial terms, first differs from A308168 in having 209.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Francisco Javier de Vega, Some Variants of Integer Multiplication, Axioms (2023) Vol. 12, 905. See p. 8.

Cf. A000009, A018818, A032742, A033630, A102627, A326836, A326837, A326841, A326850, A326851, A327783.

Select[Range[100],#==1||#-Divisors[#][[-2]]>Total[Range[Divisors[#][[-2]]-1]]&]

A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 15, 16, 17, 24, 31, 32, 36, 42, 63, 64, 69, 70, 81, 88, 98, 104, 127, 128, 136, 170, 255, 256, 277, 278, 282, 292, 325, 326, 337, 344, 354, 360, 394, 418, 424, 511, 512, 513, 514, 515, 528, 547, 561, 568, 640, 682, 768, 769, 785, 792, 896

Offset: 1

Gus Wiseman, Oct 16 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  24: (1,4)
  31: (1,1,1,1,1)
  32: (6)
  36: (3,3)
  42: (2,2,2)
  63: (1,1,1,1,1,1)
  64: (7)
  69: (4,2,1)

For regular mean we have A096199, counted by A271654 (partitions A067538).
Subsets whose geometric mean is an integer are counted by A326027.
The unordered version (partitions) is A326623, counted by A067539.
The strict case is counted by A339452, partitions A326625.
These compositions are counted by A357710.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A102627, A301987, A326622, A326624, A326028, A326567/A326568, A326641, A326645, A335405, A357184.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],IntegerQ[GeometricMean[stc[#]]]&]

A358331 Number of integer partitions of n with arithmetic and geometric mean differing by one.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 3, 3, 0, 0, 2, 2, 0, 4, 0, 0, 5, 0, 0, 4, 5, 4, 3, 2, 0, 3, 3, 10, 4, 0, 0, 7, 0, 0, 16, 2, 4, 4, 0, 0, 5, 24, 0, 6, 0, 0, 9, 0, 27, 10, 0, 7, 7, 1, 0, 44

Offset: 0

Gus Wiseman, Nov 09 2022

nonn

The arithmetic and geometric mean from such partition is a positive integer. - David A. Corneth, Nov 11 2022

			The a(30) = 2 through a(36) = 3 partitions (C = 12, G = 16):
  (888222)      .  (99333311)  (G2222222111)  .  (C9662)    (G884)
  (8844111111)                                   (C9833)    (888222111111)
                                                 (8884421)  (G42222221111)

Chai Wah Wu, Table of n, a(n) for n = 0..143
Wikipedia, Geometric mean

The version for subsets seems to be close to A178832.
These partitions are ranked by A358332.
A000041 counts partitions.
A067538 counts partitions with integer average, ranked by A316413.
A067539 counts partitions with integer geometric mean, ranked by A326623.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A078174, A102627, A271654, A326027, A326624, A326625, A326028, A326641, A326645, A357710.

Table[Length[Select[IntegerPartitions[n],Mean[#]==1+GeometricMean[#]&]],{n,0,30}]

a(n) = if (n, my(nb=0,vp); forpart(p=n, vp=Vec(p); if (vecsum(vp)/#p == 1 + sqrtn(vecprod(vp), #p), nb++)); nb, 0); \\ Michel Marcus, Nov 11 2022

from math import prod
from sympy import divisors, integer_nthroot
from sympy.utilities.iterables import partitions
def A358331(n):
    divs = {d:n//d-1 for d in divisors(n,generator=True)}
    return sum(1 for s,p in partitions(n,m=max(divs,default=0),size=True) if s in divs and (t:=integer_nthroot(prod(a**b for a, b in p.items()),s))[1] and divs[s]==t[0]) # Chai Wah Wu, Sep 24 2023

a(61)-a(80) from Giorgos Kalogeropoulos, Nov 11 2022

a(81)-a(84) from Chai Wah Wu, Sep 24 2023

A361391 Number of strict integer partitions of n with non-integer mean.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 4, 2, 4, 5, 11, 0, 17, 15, 13, 15, 37, 18, 53, 24, 48, 78, 103, 23, 111, 152, 143, 123, 255, 110, 339, 238, 372, 495, 377, 243, 759, 845, 873, 414, 1259, 842, 1609, 1383, 1225, 2281, 2589, 1285, 2827, 2518, 3904, 3836, 5119, 3715, 4630

Offset: 0

Gus Wiseman, Mar 11 2023

nonn

Are 1, 2, 4, 6, 12 the only zeros?

			The a(3) = 1 through a(11) = 11 partitions:
  {2,1}  .  {3,2}  .  {4,3}    {4,3,1}  {5,4}  {5,3,2}    {6,5}
            {4,1}     {5,2}    {5,2,1}  {6,3}  {5,4,1}    {7,4}
                      {6,1}             {7,2}  {6,3,1}    {8,3}
                      {4,2,1}           {8,1}  {7,2,1}    {9,2}
                                               {4,3,2,1}  {10,1}
                                                          {5,4,2}
                                                          {6,3,2}
                                                          {6,4,1}
                                                          {7,3,1}
                                                          {8,2,1}
                                                          {5,3,2,1}

The strict complement is counted by A102627.
The non-strict version is ranked by A348551, complement A316413.
The non-strict version is counted by A349156, complement A067538.
For median instead of mean we have A360952, complement A359907.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A307683 counts partitions with non-integer median, ranks A359912.
A325347 counts partitions with integer median, ranks A359908.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A327472 counts partitions not containing their mean, complement of A237984.
A327475 counts subsets with integer mean.
Cf. A051293, A082550, A143773, A175397, A175761, A240219, A240850, A326027, A326641, A326849, A359897.

a:= proc(m) option remember; local b; b:=
      proc(n, i, t) option remember; `if`(i*(i+1)/2Alois P. Heinz, Mar 16 2023

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!IntegerQ[Mean[#]]&]],{n,0,30}]

a(31)-a(55) from Alois P. Heinz, Mar 16 2023

A361656 Number of odd-length integer partitions of n with integer mean.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 4, 2, 1, 9, 8, 2, 13, 2, 16, 51, 1, 2, 58, 2, 85, 144, 57, 2, 49, 194, 102, 381, 437, 2, 629, 2, 1, 956, 298, 2043, 1954, 2, 491, 2293, 1116, 2, 4479, 2, 6752, 14671, 1256, 2, 193, 8035, 4570, 11614, 22143, 2, 28585, 39810, 16476, 24691, 4566

Offset: 0

Gus Wiseman, Mar 24 2023

nonn

These are partitions of n whose length is an odd divisor of n.

			The a(1) = 1 through a(10) = 8 partitions (A = 10):
  1   2   3     4   5       6     7         8   9           A
          111       11111   222   1111111       333         22222
                            321                 432         32221
                            411                 441         33211
                                                522         42211
                                                531         43111
                                                621         52111
                                                711         61111
                                                111111111
For example, the partition (3,3,2,1,1) has length 5 and mean 2, so is counted under a(10).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Odd-length partitions are counted by A027193, bisection A236559.
Including even-length gives A067538 bisected, strict A102627, ranks A316413.
The even-length version is counted by A361655.
A000041 counts integer partitions, strict A000009.
A027187 counts even-length partitions, bisection A236913.
A051293 counts subsets with integer mean, median A000975.
A058398 counts partitions by mean, see also A008284, A327482.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A326622 counts factorizations with integer mean.
Cf. A000016, A067659, A082550, A237984, A240219, A327475, A348551, A361653.

Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[Mean[#]]&]],{n,0,30}]

a(n)=if(n==0, 0, sumdiv(n, d, if(d%2, polcoef(1/prod(k=1, d, 1 - x^k + O(x^(n-d+1))), n-d)))) \\ Andrew Howroyd, Mar 24 2023

A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

Original entry on oeis.org

2, 257, 8519971, 36574494881, 140739702949921, 140773995710729, 140774004099109

Offset: 1

Gus Wiseman, Sep 27 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Conjecture: This sequence is infinite.

			The initial terms together with their binary indices:
                2: {2}
              257: {1,9}
          8519971: {1,2,6,9,18,24}
      36574494881: {1,6,8,16,18,27,32,36}
  140739702949921: {1,6,12,27,32,48}
  140773995710729: {1,4,9,12,18,32,36,48}
  140774004099109: {1,3,6,12,18,24,32,36,48}

Wikipedia, Geometric mean

A subset of A327368.
The binary weight of prime(n) is A014499(n), with binary length A035100(n).
Heinz numbers of partitions with integer mean: A316413.
Heinz numbers of partitions with integer geometric mean: A326623.
Heinz numbers with both: A326645.
Subsets with integer mean: A051293
Subsets with integer geometric mean: A326027
Subsets with both: A326643
Partitions with integer mean: A067538
Partitions with integer geometric mean: A067539
Partitions with both: A326641
Strict partitions with integer mean: A102627
Strict partitions with integer geometric mean: A326625
Strict partitions with both: A326029
Factorizations with integer mean: A326622
Factorizations with integer geometric mean: A326028
Factorizations with both: A326647
Numbers whose binary indices have integer mean: A326669
Numbers whose binary indices have integer geometric mean: A326673
Numbers whose binary indices have both: A327368
Cf. A000120, A029931, A034797, A048793, A070939, A096111, A291166, A326031, A326644, A326699/A326700.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Prime[Range[1000]],IntegerQ[Mean[bpe[#]]]&&IntegerQ[GeometricMean[bpe[#]]]&]

a(4)-a(7) from Giovanni Resta, Dec 01 2019

A339175 Number of compositions (ordered partitions) of n into distinct parts such that the average of the parts is an integer.

Original entry on oeis.org

1, 1, 1, 3, 1, 11, 1, 7, 19, 9, 1, 101, 1, 13, 193, 231, 1, 131, 1, 1411, 163, 21, 1, 3533, 3601, 25, 289, 7083, 1, 29195, 1, 3295, 451, 33, 98641, 84725, 1, 37, 649, 254007, 1, 768323, 1, 9907, 444643, 45, 1, 3420605, 2197441, 133849, 1153, 17331, 1, 12132635, 209641

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(8) = 7 because we have [8], [7, 1], [1, 7], [6, 2], [2, 6], [5, 3] and [3, 5].

Eric W. Weisstein's World of Mathematics, Arithmetic Mean
Index entries for sequences related to compositions

Cf. A032020, A067538, A102627, A339452.

A360670 Number of integer partitions of n whose parts have the same mean as their negated first differences.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 5, 1, 0, 3, 3, 0, 4, 0, 5, 3, 0, 2, 10, 0, 0, 4, 10, 0, 5, 0, 7, 9, 0, 0, 17, 1, 6, 5, 10, 0, 9, 8, 14, 6, 0, 0, 34, 0, 0, 9, 18, 13, 13, 0, 17, 7, 19, 0, 40, 0, 0, 28

Offset: 0

Gus Wiseman, Feb 18 2023

nonn

			The a(n) partitions for n = 4, 12, 24, 27, 30, 44:
  (3,1)  (9,3)    (18,6)      (19,7,1)  (21,8,1)      (33,11)
         (9,2,1)  (17,6,1)    (20,5,2)  (22,6,2)      (34,5,4,1)
                  (18,4,2)    (21,3,3)  (23,4,3)      (34,6,3,1)
                  (19,2,2,1)            (25,2,1,1,1)  (34,7,2,1)
                  (19,3,1,1)                          (34,8,1,1)
                                                      (35,4,3,2)
                                                      (35,5,2,2)
For example, the partition y = (28,4,3,1), with mean 9, has negated first differences (24,2,1), with mean 9, so y is counted under a(36).

These partitions have ranks A360669.
A000041 counts integer partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A326567/A326568 gives mean of prime indices.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A060016, A240219, A116608, A316313, A340610, A360068.

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

A360683 Number of integer partitions of n whose second differences sum to 0, meaning either there is only one part, or the first two parts have the same difference as the last two parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 6, 11, 12, 17, 14, 32, 23, 40, 44, 64, 59, 104, 93, 149, 157, 218, 227, 342, 349, 481, 538, 713, 777, 1052, 1145, 1494, 1692, 2130, 2416, 3064, 3449, 4286, 4918, 6028, 6882, 8424, 9620, 11634, 13396, 16022, 18416, 22019, 25248, 29954

Offset: 0

Gus Wiseman, Feb 19 2023

nonn

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (22111)    (71)
                                     (321)     (1111111)  (2222)
                                     (2211)               (3221)
                                     (111111)             (3311)
                                                          (22211)
                                                          (221111)
                                                          (11111111)

For mean instead of sum we have a(n) - A008619(n).
For median instead of sum we have A360682.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A067538 counts partitions with integer mean, strict A102627.
A316413 ranks partitions with integer mean, complement A348551.
Cf. A114638, A240219, A325347, A360068.

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

A364060 Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1, 0, 2, 4, 0, 0, 1, 0, 2, 5, 3, 0, 0, 1, 0, 4, 7, 0, 3, 0, 0, 1, 0, 4, 8, 5, 4, 0, 0, 0, 1, 0, 4, 14, 7, 4, 0, 0, 0, 0, 1, 0, 7, 21, 8, 0, 5, 0, 0, 0, 0, 1, 0, 7, 22, 11, 10, 0, 5, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Jul 07 2023

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

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  2  0  1
  0  2  4  0  0  1
  0  2  5  3  0  0  1
  0  4  7  0  3  0  0  1
  0  4  8  5  4  0  0  0  1
  0  4 14  7  4  0  0  0  0  1
  0  7 21  8  0  5  0  0  0  0  1
  0  7 22 11 10  0  5  0  0  0  0  1
  0  7 36 15 12  0  6  0  0  0  0  0  1
  0 12 32 36 14  0  6  0  0  0  0  0  0  1
  0 12 53 23 23 16  0  7  0  0  0  0  0  0  1
  0 12 80 30 27 19  0  0  7  0  0  0  0  0  0  1
Row n = 7 counts the following partitions:
  .  (31111)    (511)   .  (61)  .  .  (7)
     (22111)    (421)      (52)
     (211111)   (4111)     (43)
     (1111111)  (331)
                (322)
                (3211)
                (2221)

Wikipedia, Rounding.

Row sums are A000041.
The rank statistic for this triangle is A363489.
The version for low mean is A363945, rank statistic A363943.
The version for high mean is A363946, rank statistic A363944.
Column k = 1 is A363947 (A026905 tripled).
A008284 counts partitions by length, A058398 by mean.
A026905 redoubled counts partitions with high mean 2, ranks A363950.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
More triangles: A124943, A124944, A363952, A363953.
Cf. A002865, A025065, A237984, A327472, A327482, A363723, A363724, A363731.

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

cp's OEIS Frontend

A327782 Numbers n that cannot be written as a sum of two or more distinct parts with the largest part dividing n.

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A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

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A358331 Number of integer partitions of n with arithmetic and geometric mean differing by one.

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A361391 Number of strict integer partitions of n with non-integer mean.

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Maple

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A361656 Number of odd-length integer partitions of n with integer mean.

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PARI

A327777 Prime numbers whose binary indices have integer mean and integer geometric mean.

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Mathematica

Extensions

A339175 Number of compositions (ordered partitions) of n into distinct parts such that the average of the parts is an integer.

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A360670 Number of integer partitions of n whose parts have the same mean as their negated first differences.

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Mathematica