A050314 -id:A050314 - cp's OEIS frontend

A050315 Main diagonal of A050314.

1, 1, 1, 2, 1, 2, 2, 5, 1, 2, 2, 5, 2, 5, 5, 15, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15, 15, 52, 5, 15, 15, 52, 15, 52, 52, 203, 1, 2, 2, 5, 2, 5, 5, 15, 2, 5, 5, 15, 5, 15, 15, 52, 2, 5, 5, 15, 5, 15

Offset: 0

Christian G. Bower, Sep 15 1999

nonn

Also, a(n) is the number of odd multinomial coefficients n!/(k_1!...k_m!) with 1 <= k_1 <= ... <= k_m and k_1 + ... + k_m = n. - Pontus von Brömssen, Mar 23 2018
From Gus Wiseman, Mar 30 2019: (Start)
Also the number of strict integer partitions of n with no binary carries. The Heinz numbers of these partitions are given by A325100. A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion. For example, the a(1) = 1 through a(15) = 15 strict integer partitions with no binary carries are:
(1) (2) (3)  (4) (5)  (6)  (7)   (8) (9)  (A)  (B)   (C)  (D)   (E)   (F)
        (21)     (41) (42) (43)      (81) (82) (83)  (84) (85)  (86)  (87)
                           (52)                (92)       (94)  (A4)  (96)
                           (61)                (A1)       (C1)  (C2)  (A5)
                           (421)               (821)      (841) (842) (B4)
                                                                      (C3)
                                                                      (D2)
                                                                      (E1)
                                                                      (843)
                                                                      (852)
                                                                      (861)
                                                                      (942)
                                                                      (A41)
                                                                      (C21)
                                                                      (8421)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..16383
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000110, A000120, A050314.
Cf. A070939, A080572, A247935, A267610.
Cf. A325093, A325095, A325096, A325099, A325100, A325103, A325110, A325123.
Main diagonal of A307431 and of A307505.

a:= n-> combinat[bell](add(i,i=convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 08 2019

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&stableQ[#,Intersection[binpos[#1],binpos[#2]]!={}&]&]],{n,0,20}] (* Gus Wiseman, Mar 30 2019 *)
a[n_] := BellB[DigitCount[n, 2, 1]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021 *)

Bell number of number of 1's in binary: a(n) = A000110(A000120(n)).

A050316 a(n) = A050314(2n+1,1): column 1 of triangle.

Original entry on oeis.org

1, 1, 3, 4, 8, 11, 23, 35, 54, 79, 123, 177, 260, 368, 557, 831, 1160, 1677, 2346, 3310, 4545, 6326, 8611, 11799, 15805, 21344, 28352, 37835, 49701, 65730, 86123, 113520, 147322, 193012, 249649, 324677, 418017, 541731, 695399, 894747, 1143262, 1464785, 1864629

Offset: 0

Christian G. Bower, Sep 15 1999

nonn

Cf. A050314.

A048833 Number of starting positions of Nim with 2n pieces such that 2nd player wins. Partitions of 2n such that xor-sum of partitions is 0.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 16, 31, 43, 68, 98, 153, 213, 317, 443, 704, 971, 1415, 1975, 2818, 3865, 5401, 7366, 10142, 13639, 18438, 24583, 32861, 43345, 57268, 75175, 99119, 129278, 168796, 219614, 284887, 368546, 475919, 614379, 788845, 1012117, 1293980, 1654090

Offset: 0

Christian G. Bower, Jun 15 1999

nonn

Number of different prime signatures of the 2n-almost primes in A268390. - Peter Munn, Dec 02 2021

			For n=4 the 6 partitions of 8 are [1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2, 2], [2, 2, 2, 2], [1, 1, 1, 2, 3], [1, 1, 3, 3] and [4, 4].

Alois P. Heinz, Table of n, a(n) for n = 0..750
C. L. Bouton, Nim, a game with a complete mathematical theory, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.
R. J. Nowakowski, G. Renault, E. Lamoureux, S. Mellon and T. Miller, The Game of timber!, hal-00985731, 2013.

Cf. A050314.

Cf. A003987, A268390.

read("transforms") : # defines XORnos
A048833 := proc(n)
    local p, xrs,i,a ;
    if n = 0 then
        return 1 ;
    end if;
    a := 0 ;
    for p in combinat[partition](2*n) do
        xrs := op(1,p) ;
        for i from 2 to nops(p) do
            xrs := XORnos(xrs,op(i,p)) ;
        end do:
        if xrs = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 29 2022

b[n_, i_, k_] := b[n, i, k] = If[n == 0, x^k, If[i < 1, 0, Sum[b[n-i*j, i-1, If[EvenQ[j], k, BitXor[i, k]]], {j, 0, n/i}]]];
a[n_] := Coefficient[b[2n, 2n, 0], x, 0];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Mar 25 2024, after Alois P. Heinz in A050314 *)

a(n) = A050314(2n, 0): column 0 of triangle.

A235488 Squarefree numbers which yield zero when their prime factors are xored together.

Original entry on oeis.org

70, 646, 1798, 2145, 3526, 5865, 6006, 9177, 11305, 13110, 16422, 20553, 20806, 21489, 23529, 28905, 28985, 30305, 31465, 37961, 38086, 38454, 42441, 44022, 44998, 45353, 45942, 46345, 53985, 54230, 55913, 60630, 60697, 61705, 62049, 64790, 78406, 80934, 81158

Offset: 1

Antti Karttunen, Jan 22 2014

All n for which A008683(n) <> 0 and A072594(n) = 0.
It seems that an analogous case as A072595 for GF(2)[X]-polynomials is just the squares of GF(2)[X]-polynomials (A000695), thus in that ring, the sequence analogous to this one would be empty.
This sequence happens also to encode in the prime factorization of n a certain subset of the Nim game positions that are second-player win.

			70 is included, as 70 = 2*5*7, whose binary representations are '10', '101' and '111', which when all are xored (cf. A003987) together, cancel all 1-bits, thus yielding zero.
212585 is included, as 212585 = 5*17*41*61, and when we xor their base-2 representations together:
     101
   10001
  101001
  111101
--------
  000000
we get only zeros, because in each column (bit-position), there is an even number of 1-bits.

Antti Karttunen and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 75 terms from Karttunen)
Wikipedia, Nim

Intersection of A005117 and A072595 (equally: of A005117 and A072596).

Cf. A003987, A235490, A079599, A048833, A050314.

Select[Range[82000],SquareFreeQ[#]&&BitXor@@FactorInteger[#][[All,1]]==0&] (* Harvey P. Dale, Apr 01 2017 *)

is(n)=if(n<9, return(0)); my(f=factor(n)); vecmax(f[,2])==1 && fold(bitxor, f[,1])==0 \\ Charles R Greathouse IV, Aug 06 2016

A307431 Number T(n,k) of partitions of n into parts whose bitwise OR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 1, 0, 1, 0, 4, 0, 2, 0, 1, 1, 5, 0, 2, 2, 0, 1, 0, 7, 0, 2, 0, 5, 0, 1, 1, 8, 1, 2, 2, 6, 1, 0, 1, 0, 11, 0, 4, 0, 12, 0, 2, 0, 1, 1, 12, 0, 5, 4, 15, 0, 2, 2, 0, 1, 0, 15, 0, 5, 0, 28, 0, 2, 0, 5, 0, 1, 1, 17, 1, 5, 5, 35, 0, 2, 2, 6, 2

Offset: 0

Alois P. Heinz, Apr 08 2019

			T(6,1) = 1: 111111.
T(6,2) = 1: 222.
T(6,3) = 5: 11112, 1122, 1113, 123, 33.
T(6,5) = 2: 114, 15.
T(6,6) = 2: 24, 6.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0,  2;
  0, 1, 1,  2, 1;
  0, 1, 0,  4, 0, 2;
  0, 1, 1,  5, 0, 2, 2;
  0, 1, 0,  7, 0, 2, 0,  5;
  0, 1, 1,  8, 1, 2, 2,  6, 1;
  0, 1, 0, 11, 0, 4, 0, 12, 0, 2;
  0, 1, 1, 12, 0, 5, 4, 15, 0, 2, 2;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Bitwise operation
Wikipedia, Partition (number theory)

Columns k=0-1 give: A000007, A057427.
Row sums give: A000041.
Main diagonal gives A050315.
Cf. A050314 (the same for XOR), A307432 (the same for AND).

b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i), Bits[Or](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..14);

A307432 Number T(n,k) of partitions of n into parts whose bitwise AND equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 3, 2, 1, 0, 0, 1, 5, 3, 1, 1, 0, 0, 1, 9, 4, 1, 0, 0, 0, 0, 1, 11, 6, 3, 0, 1, 0, 0, 0, 1, 18, 6, 3, 1, 1, 0, 0, 0, 0, 1, 27, 8, 3, 1, 1, 1, 0, 0, 0, 0, 1, 38, 11, 4, 0, 2, 0, 0, 0, 0, 0, 0, 1, 53, 13, 6, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, Apr 08 2019

			T(6,0) = 5: 11112, 1122, 123, 114, 24.
T(6,1) = 3: 111111, 1113, 15.
T(6,2) = 1: 222.
T(6,3) = 1: 33.
T(6,6) = 1: 6.
Triangle T(n,k) begins:
   1;
   0, 1;
   0, 1, 1;
   1, 1, 0, 1;
   1, 2, 1, 0, 1;
   3, 2, 1, 0, 0, 1;
   5, 3, 1, 1, 0, 0, 1;
   9, 4, 1, 0, 0, 0, 0, 1;
  11, 6, 3, 0, 1, 0, 0, 0, 1;
  18, 6, 3, 1, 1, 0, 0, 0, 0, 1;
  27, 8, 3, 1, 1, 1, 0, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Bitwise operation
Wikipedia, Partition (number theory)

Column k=0 gives A307435.
Row sums give A000041.
Main diagonal gives A000012.
Cf. A050314 (the same for XOR), A307431 (the same for OR).

b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i), Bits[And](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(
         b(n$2, `if`(n=0, 0, 2^ilog2(2*n)-1))):
seq(T(n), n=0..14);

A233810 Number of starting configurations of Nim with n pieces such that 1st player wins. Partitions of n such that their xor-sum is nonzero.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 7, 15, 16, 30, 32, 56, 61, 101, 104, 176, 188, 297, 317, 490, 529, 792, 849, 1255, 1362, 1958, 2119, 3010, 3275, 4565, 4900, 6842, 7378, 10143, 10895, 14883, 16002, 21637, 23197, 31185, 33473, 44583, 47773, 63261, 67809, 89134, 95416, 124754, 133634, 173525, 185788, 239943, 257006, 329931, 353294, 451276, 483478, 614154, 657952, 831820, 891292, 1121505, 1201037, 1505499, 1612352, 2012558, 2154724, 2679689, 2868121, 3554345, 3803081, 4697205, 5024237, 6185689, 6613581, 8118264, 8674712, 10619863, 11343319, 13848650, 14784359, 18004327

Offset: 0

Álvar Ibeas, Dec 16 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1500
C. L. Bouton, Nim, a game with a complete mathematical theory, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.

Cf. A000041, A048833, A050314.

b[n_, i_, k_] := b[n, i, k] = If[n == 0, x^k, If[i<1, 0, Sum[b[n-i*j, i-1, If[EvenQ[j], k, BitXor[i, k]]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]]; a[n_] := Total[Rest[T[n]]]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Nov 14 2016, after Alois P. Heinz *)

a(n) = Sum_{k>0} A050314(n,k). [Row sums of A050314 minus the leftmost term on each row]

a(2n+1) = A000041(2n+1), a(2n) = A000041(2n)-A048833(n).

A307505 Number T(n,k) of partitions of n into distinct parts whose bitwise XOR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 2, 1, 0, 1, 0, 5, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 1, 0, 0, 0, 5, 1, 0, 5, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2

Offset: 0

Alois P. Heinz, Apr 11 2019

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 2;
  0, 0, 1, 0, 1;
  0, 1, 0, 0, 0, 2;
  1, 0, 0, 0, 1, 0, 2;
  0, 0, 0, 0, 0, 0, 0, 5;
  0, 0, 0, 0, 1, 0, 4, 0, 1;
  0, 1, 0, 0, 0, 4, 0, 1, 0, 2;
  1, 0, 1, 0, 5, 0, 0, 0, 1, 0, 2;
  ...

Alois P. Heinz, Rows n = 0..360, flattened
Wikipedia, Bitwise operation
Wikipedia, Partition (number theory)

Bisection (even part) of column k=0 gives A307506.
Row sums give A000009.
Main diagonal gives A050315.
Cf. A050314.

b:= proc(n, i, k) option remember; `if`(n=0, x^k, `if`(i<1, 0,
      b(n, i-1, k)+b(n-i, min(n-i, i-1), Bits[Xor](i, k))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..14);

T(n,k) = 0 if n+k is odd.

A370874 Number of partitions of 4n whose xor-sum is 2n.

Original entry on oeis.org

1, 2, 4, 16, 16, 65, 153, 411, 165, 437, 931, 2317, 4802, 10595, 21565, 43211, 5014, 10911, 22466, 44695, 83058, 156147, 286432, 516479, 595305, 1133892, 2111273, 3803940, 6731760, 11653790, 19886537, 33275225, 916662, 1593595, 2753582, 4676617, 7866137

Offset: 0

Alois P. Heinz, Mar 25 2024

nonn

			a(0) = 1: the empty partition.
a(1) = 2: 211, 31.
a(2) = 4: 41111, 422, 5111, 62.
a(3) = 16: 42111111, 422211, 4311111, 43221, 4332, 5211111, 52221, 531111, 5322, 6111111, 62211, 6321, 633, 711111, 7221, 732.
a(4) = 16: 811111111, 8221111, 82222, 832111, 83311, 844, 91111111, 922111, 93211, 9331, (10)21111, (10)222, (10)3111, (11)2111, (11)311, (12)4.

Alois P. Heinz, Table of n, a(n) for n = 0..511
Wikipedia, Partition (number theory)

Cf. A000041, A050314, A058696.

b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
     `if`(i<1 or ilog2(k)>ilog2(i), 0, b(n, i-1, k)+
        b(n-i, min(n-i,i), Bits[Xor](i, k))))
    end:
a:= n-> b(4*n$2, 2*n):
seq(a(n), n=0..36);

a(n) = A050314(4n,2n).

cp's OEIS Frontend

A050315 Main diagonal of A050314.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A050316 a(n) = A050314(2n+1,1): column 1 of triangle.

Views

Author

Keywords

Crossrefs

A048833 Number of starting positions of Nim with 2n pieces such that 2nd player wins. Partitions of 2n such that xor-sum of partitions is 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A235488 Squarefree numbers which yield zero when their prime factors are xored together.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A307431 Number T(n,k) of partitions of n into parts whose bitwise OR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A307432 Number T(n,k) of partitions of n into parts whose bitwise AND equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A233810 Number of starting configurations of Nim with n pieces such that 1st player wins. Partitions of n such that their xor-sum is nonzero.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A307505 Number T(n,k) of partitions of n into distinct parts whose bitwise XOR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A370874 Number of partitions of 4n whose xor-sum is 2n.

Views

Author