A327471 -id:A327471 - cp's OEIS frontend

A327476 Heinz numbers of integer partitions whose mean A326567/A326568 is not a part.

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106

Offset: 1

Gus Wiseman, Sep 13 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}

Complement of A327473.
The enumeration of these partitions by sum is given by A327472.
Subsets whose mean is not an element are A327471.
Cf. A056239, A067538, A112798, A114639, A237984, A240851, A316413, A324756, A324758, A326567/A326568, A327477.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MemberQ[primeMS[#],Mean[primeMS[#]]]&]

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94, 172, 316, 586, 1096, 2048, 3856, 7286, 13798, 26216, 49940, 95326, 182362, 349536, 671092, 1290556, 2485534, 4793492, 9256396, 17895736, 34636834, 67108864, 130150588, 252645136, 490853416

Offset: 0

N. J. A. Sloane

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.
Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).
Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015
Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016
From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:
  1  2  3    4    5      6      7        8
        123  234  135    246    147      258
                  345    456    357      468
                  12345  1236   567      678
                         1456   2347     1348
                         23456  2567     1568
                                12467    3458
                                13457    3678
                                34567    12458
                                1234567  14578
                                         23578
                                         24568
                                         45678
                                         123468
                                         135678
                                         2345678
(End)
Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

			For n=3 the 2 output sequences are 000111000111... and 010101...
For n=5 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
For n=6 there are 6 such sequences.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
J. Hedetniemi and K. R. Hutson, Equilibrium of shortest path load in ring network, Congressus Numerant., 203 (2010), 75-95. See p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Diff. Eqs. 20 (1) (2008) 201, eq. (39)

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
A. E. Brouwer, The Enumeration of Locally Transitive Tournaments, Math. Centr. Report ZW138, Amsterdam, 1980.
S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, Estimating the size of correcting codes using extremal graph problems, Optimization, 227-243, Springer Optim. Appl., 32, Springer, New York, 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Sébastien Designolle, Tamás Vértesi, and Sebastian Pokutta, Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms, arXiv:2310.20677 [quant-ph], 2023.
T. M. A. Fink, Exact dynamics of the critical Kauffman model with connectivity one, arXiv:2302.05314 [cond-mat.stat-mech], 2023.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, Challenge Problems: Independent Sets in Graphs
Yan Bo Ti, Gabriel Verret, and Lukas Zobernig, Abelian Varieties with p-rank Zero, arXiv:2203.08401 [math.NT], 2022.
Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra Appl. 530, 414-444 (2017).
Index entries for sequences related to tournaments
Index entries for sequences related to necklaces
Index entries for sequences related to subset sums modulo m

The main diagonal of table A068009, the left edge of triangle A053633.
Cf. A000048, A000031, A000013, A053634, A182469.
Subsets whose mean is an element are A065795.
Dominated by A082550.
Partitions containing their mean are A237984.
Subsets containing n but not their mean are A327477.
Cf. A051293, A135342, A240850, A327471, A327478.

a000016 0 = 1
a000016 n = (`div` (2 * n)) $ sum $
   zipWith (*) (map a000010 oddDivs) (map ((2 ^) . (div n)) $ oddDivs)
   where oddDivs = a182469_row n
-- Reinhard Zumkeller, May 01 2012

A000016 := proc(n) local d, t; if n = 0 then return 1 else t := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t := t + NumberTheory:-Totient(d)* 2^(n/d)/(2*n) fi od; return t fi end:

a[0] = 1; a[n_] := Sum[Mod[k, 2] EulerPhi[k]*2^(n/k)/(2*n), {k, Divisors[n]}]; Table[a[n], {n, 0, 35}](* Jean-François Alcover, Feb 17 2012, after Pari *)

a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n));

from sympy import totient, divisors
def A000016(n): return sum(totient(d)<>(~n&n-1).bit_length(),generator=True))//n if n else 1 # Chai Wah Wu, Feb 21 2023

a(n) = Sum_{odd d divides n} (phi(d)*2^(n/d))/(2*n), n>0.
a(n) = A063776(n)/2.
a(n) = 2^(n-1) - A327477(n). - Gus Wiseman, Sep 14 2019

More terms from Michael Somos, Dec 11 1999

A327472 Number of integer partitions of n not containing their mean.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 6, 13, 16, 25, 34, 54, 56, 99, 121, 154, 201, 295, 324, 488, 541, 725, 957, 1253, 1292, 1892, 2356, 2813, 3378, 4563, 4838, 6840, 7686, 9600, 12076, 14180, 15445, 21635, 25627, 29790, 33309, 44581, 48486, 63259, 70699, 82102, 104553, 124752

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(3) = 1 through a(8) = 16 partitions not containing their mean:
  (21)  (31)   (32)    (42)     (43)      (53)
        (211)  (41)    (51)     (52)      (62)
               (221)   (411)    (61)      (71)
               (311)   (2211)   (322)     (332)
               (2111)  (3111)   (331)     (422)
                       (21111)  (421)     (431)
                                (511)     (521)
                                (2221)    (611)
                                (3211)    (3311)
                                (4111)    (5111)
                                (22111)   (22211)
                                (31111)   (32111)
                                (211111)  (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

The Heinz numbers of these partitions are A327476.
Partitions with their mean are A237984.
Subsets without their mean are A327471.
Subsets with n but without their mean are A327477.
Strict partitions without their mean are A240851.
Cf. A007865, A067538, A067538, A102627, A114639, A324756, A327473.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,Mean[#]]&]],{n,0,20}]

from sympy.utilities.iterables import partitions
def A327472(n): return sum(1 for s,p in partitions(n,size=True) if n%s or n//s not in p) if n else 1 # Chai Wah Wu, Sep 21 2023

A065795 Number of subsets of {1,2,...,n} that contain the average of their elements.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 26, 42, 72, 124, 218, 390, 706, 1292, 2388, 4436, 8292, 15578, 29376, 55592, 105532, 200858, 383220, 732756, 1403848, 2694404, 5179938, 9973430, 19229826, 37125562, 71762396, 138871260, 269021848, 521666984, 1012520400, 1966957692, 3824240848

Offset: 1

John W. Layman, Dec 05 2001

nonn

Also the number of subsets of {1,2,...,n} with sum of entries divisible by the largest element (compare A000016). See the Palmer Melbane link for a bijection. - Joel B. Lewis, Nov 13 2014

			a(4)=6, since {1}, {2}, {3}, {4}, {1,2,3} and {2,3,4} contain their averages.
From _Gus Wiseman_, Sep 14 2019: (Start)
The a(1) = 1 through a(6) = 16 subsets:
  {1}  {1}  {1}      {1}      {1}          {1}
       {2}  {2}      {2}      {2}          {2}
            {3}      {3}      {3}          {3}
            {1,2,3}  {4}      {4}          {4}
                     {1,2,3}  {5}          {5}
                     {2,3,4}  {1,2,3}      {6}
                              {1,3,5}      {1,2,3}
                              {2,3,4}      {1,3,5}
                              {3,4,5}      {2,3,4}
                              {1,2,3,4,5}  {2,4,6}
                                           {3,4,5}
                                           {4,5,6}
                                           {1,2,3,6}
                                           {1,4,5,6}
                                           {1,2,3,4,5}
                                           {2,3,4,5,6}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..3333
Palmer Melbane, Combinations under constraint, Art of Problem Solving thread. - _Joel B. Lewis_, Nov 13 2014

Subsets containing n whose mean is an element are A000016.
The version for integer partitions is A237984.
Subsets not containing their mean are A327471.
Cf. A051293, A063776, A066571, A135342, A324736, A325705, A326083, A326836, A327478, A327481.

Table[ Sum[a = Select[Divisors[i], OddQ[ # ] &]; Apply[ Plus, 2^(i/a) * EulerPhi[a]]/i, {i, n}]/2, {n, 34}]
(* second program *)
Table[Length[Select[Subsets[Range[n]],MemberQ[#,Mean[#]]&]],{n,0,10}] (* Gus Wiseman, Sep 14 2019 *)

a(n) = (1/2)*sum(i=1, n, (1/i)*sumdiv(i, d, if (d%2, 2^(i/d)*eulerphi(d)))); \\ Michel Marcus, Dec 20 2020

from sympy import totient, divisors
def A065795(n): return sum((sum(totient(d)<>(~k&k-1).bit_length(),generator=True))<<1)//k for k in range(1,n+1))>>1 # Chai Wah Wu, Feb 22 2023

a(n) = (1/2)*Sum_{i=1..n} (f(i) - 1) where f(i) = (1/i) * Sum_{d | i and d is odd} 2^(i/d) * phi(d).
a(n) = (n + A051293(n))/2.
a(n) = 2^n - A327471(n). - Gus Wiseman, Sep 14 2019

Edited and extended by Robert G. Wilson v, Nov 15 2002

A327477 Number of subsets of {1..n} containing n whose mean is not an element.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 26, 54, 112, 226, 460, 930, 1876, 3780, 7606, 15288, 30720, 61680, 123786, 248346, 498072, 998636, 2001826, 4011942, 8039072, 16106124, 32263876, 64623330, 129424236, 259179060, 518975176, 1039104990, 2080374784, 4164816708, 8337289456

Offset: 0

Gus Wiseman, Sep 13 2019

nonn

			The a(1) = 1 through a(5) = 12 subsets:
  {1,2}  {1,3}  {1,4}      {1,5}
         {2,3}  {2,4}      {2,5}
                {3,4}      {3,5}
                {1,2,4}    {4,5}
                {1,3,4}    {1,2,5}
                {1,2,3,4}  {1,4,5}
                           {2,3,5}
                           {2,4,5}
                           {1,2,3,5}
                           {1,2,4,5}
                           {1,3,4,5}
                           {2,3,4,5}

Subsets whose mean is an element are A065795.
Subsets whose mean is not an element are A327471.
Subsets containing n whose mean is an element are A000016.
Cf. A007865, A051293, A082550, A135342, A240851, A324741, A327472.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!MemberQ[#,Mean[#]]&]],{n,0,10}]

from sympy import totient, divisors
def A327477(n): return (1<>(~n&n-1).bit_length(),generator=True))//n if n else 0 # Chai Wah Wu, Feb 21 2023

From Alois P. Heinz, Feb 21 2023: (Start)
a(n) = A327471(n) - A327471(n-1) for n>=1.
a(n) = 2^(n-1) - A000016(n) for n>=1. (End)

a(25)-a(34) from Alois P. Heinz, Feb 21 2023

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A327476 Heinz numbers of integer partitions whose mean A326567/A326568 is not a part.

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A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

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A327472 Number of integer partitions of n not containing their mean.

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A065795 Number of subsets of {1,2,...,n} that contain the average of their elements.

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A327477 Number of subsets of {1..n} containing n whose mean is not an element.

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Python

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