A028260 -id:A028260 - cp's OEIS frontend

A340101 Number of factorizations of 2n + 1 into odd factors > 1.

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 4, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 5, 1, 1, 2, 1, 2, 4, 2, 2, 2, 3, 1, 2, 1, 2, 7, 1, 1, 2, 2, 2, 4, 1, 1, 4, 2, 1, 2, 2, 1, 5, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 2, 7, 1, 1, 5, 1, 1, 2, 2, 2, 4, 2

Offset: 0

Gus Wiseman, Dec 28 2020

nonn

			The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
  27      45      135       225       315       405         1155
  3*9     5*9     3*45      3*75      5*63      5*81        15*77
  3*3*3   3*15    5*27      5*45      7*45      9*45        21*55
          3*3*5   9*15      9*25      9*35      15*27       33*35
                  3*5*9     15*15     15*21     3*135       3*385
                  3*3*15    5*5*9     3*105     5*9*9       5*231
                  3*3*3*5   3*3*25    5*7*9     3*3*45      7*165
                            3*5*15    3*3*35    3*5*27      11*105
                            3*3*5*5   3*5*21    3*9*15      3*5*77
                                      3*7*15    3*3*5*9     3*7*55
                                      3*3*5*7   3*3*3*15    5*7*33
                                                3*3*3*3*5   3*11*35
                                                            5*11*21
                                                            7*11*15
                                                            3*5*7*11

Antti Karttunen, Table of n, a(n) for n = 0..32768

The version for partitions is A160786, ranked by A300272.
The even version is A340785.
The odd-length case is A340102.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A316439 counts factorizations by product and length.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.
Odd bisection of A001055, and also of A349907.

g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> g(2*n+1$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ[Times@@#]&]],{n,1,100,2}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s)); \\ After code in A001055
A340101(n) = A001055(n+n+1); \\ Antti Karttunen, Dec 13 2021

a(n) = A001055(2n+1).

a(n) = A349907(2n+1). - Antti Karttunen, Dec 13 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 13 2021

A345919 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum < 0.

Original entry on oeis.org

6, 12, 20, 24, 25, 27, 30, 40, 48, 49, 51, 54, 60, 72, 80, 81, 83, 86, 92, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 111, 116, 120, 121, 123, 126, 144, 160, 161, 163, 166, 172, 184, 192, 193, 194, 195, 197, 198, 199, 202, 204, 205, 207, 212, 216, 217, 219

Offset: 1

Gus Wiseman, Jul 09 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

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 initial terms and the corresponding compositions:
      6: (1,2)         81: (2,4,1)
     12: (1,3)         83: (2,3,1,1)
     20: (2,3)         86: (2,2,1,2)
     24: (1,4)         92: (2,1,1,3)
     25: (1,3,1)       96: (1,6)
     27: (1,2,1,1)     97: (1,5,1)
     30: (1,1,1,2)     98: (1,4,2)
     40: (2,4)         99: (1,4,1,1)
     48: (1,5)        101: (1,3,2,1)
     49: (1,4,1)      102: (1,3,1,2)
     51: (1,3,1,1)    103: (1,3,1,1,1)
     54: (1,2,1,2)    106: (1,2,2,2)
     60: (1,1,1,3)    108: (1,2,1,3)
     72: (3,4)        109: (1,2,1,2,1)
     80: (2,5)        111: (1,2,1,1,1,1)

The version for Heinz numbers of partitions is A119899.
These are the positions of terms < 0 in A124754.
These compositions are counted by A294175 (even bisection: A008549).
The complement is A345913.
The weak (k <= 0) version is A345915.
The opposite (k < 0) version is A345917.
The version for reversed alternating sum is A345920.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]<0&]

A340601 Number of integer partitions of n of even rank.

Original entry on oeis.org

1, 1, 0, 3, 1, 5, 3, 11, 8, 18, 16, 34, 33, 57, 59, 98, 105, 159, 179, 262, 297, 414, 478, 653, 761, 1008, 1184, 1544, 1818, 2327, 2750, 3480, 4113, 5137, 6078, 7527, 8899, 10917, 12897, 15715, 18538, 22431, 26430, 31805, 37403, 44766, 52556, 62620, 73379

Offset: 0

Gus Wiseman, Jan 21 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its number of parts. For this sequence, the rank of an empty partition is 0.

			The a(1) = 1 through a(9) = 18 partitions (empty column indicated by dot):
  (1)  .  (3)    (22)  (5)      (42)    (7)        (44)      (9)
          (21)         (41)     (321)   (43)       (62)      (63)
          (111)        (311)    (2211)  (61)       (332)     (81)
                       (2111)           (322)      (521)     (333)
                       (11111)          (331)      (2222)    (522)
                                        (511)      (4211)    (531)
                                        (2221)     (32111)   (711)
                                        (4111)     (221111)  (4221)
                                        (31111)              (4311)
                                        (211111)             (6111)
                                        (1111111)            (32211)
                                                             (33111)
                                                             (51111)
                                                             (222111)
                                                             (411111)
                                                             (3111111)
                                                             (21111111)
                                                             (111111111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: Heinz numbers are given in parentheses below.
The positive case is A101708 (A340605).
The Heinz numbers of these partitions are A340602.
The odd version is A340692 (A340603).
- Rank -
A047993 counts partitions of rank 0 (A106529).
A072233 counts partitions by sum and length.
A101198 counts partitions of rank 1 (A325233).
A101707 counts partitions of odd positive rank (A340604).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A340653 counts factorizations of rank 0.
- Even -
A024430 counts set partitions of even length.
A027187 counts partitions of even length (A028260).
A027187 (also) counts partitions of even maximum (A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A052841 counts ordered set partitions of even length.
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts even-length partitions of even numbers (A340784).
A339846 counts factorizations of even length.
Cf. A000041, A006141, A039900, A064174, A067538, A117409, A200750, A324516.

b:= proc(n, i, r) option remember; `if`(n=0, 1-max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 22 2021

Table[If[n==0,1,Length[Select[IntegerPartitions[n],EvenQ[Max[#]-Length[#]]&]]],{n,0,30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, 1 - Max[0, r], If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 - If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

p_q(k) = {prod(j=1, k, 1-q^j); }
GB_q(N, M)= {if(N>=0 && M>=0,  p_q(N+M)/(p_q(M)*p_q(N)), 0 ); }
A_q(N) = {my(q='q+O('q^N), g=1+sum(i=1,N, sum(j=1,N/i, q^(i*j) * ( ((1/2)*(1+(-1)^(i+j))) + sum(k=1,N-(i*j), ((q^k)*GB_q(k,i-2)) * ((1/2)*(1+(-1)^(i+j+k)))))))); Vec(g)}
A_q(50) \\ John Tyler Rascoe, Apr 15 2024

G.f.: 1 + Sum_{i, j>0} q^(i*j) * ( (1+(-1)^(i+j))/2 + Sum_{k>0} q^k * q_binomial(k,i-2) * (1+(-1)^(i+j+k))/2 ). - John Tyler Rascoe, Apr 15 2024

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - Vaclav Kotesovec, Apr 17 2024

A345918 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum > 0.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 11, 12, 14, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 37, 38, 40, 42, 44, 47, 48, 51, 52, 54, 56, 59, 60, 62, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88, 91, 92, 93, 94, 96, 99, 100, 101, 102, 104, 106, 107, 108

Offset: 1

Gus Wiseman, Jul 09 2021

nonn

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The initial terms and the corresponding compositions:
     1: (1)        26: (1,2,2)        52: (1,2,3)
     2: (2)        27: (1,2,1,1)      54: (1,2,1,2)
     4: (3)        28: (1,1,3)        56: (1,1,4)
     6: (1,2)      30: (1,1,1,2)      59: (1,1,2,1,1)
     7: (1,1,1)    31: (1,1,1,1,1)    60: (1,1,1,3)
     8: (4)        32: (6)            62: (1,1,1,1,2)
    11: (2,1,1)    35: (4,1,1)        64: (7)
    12: (1,3)      37: (3,2,1)        67: (5,1,1)
    14: (1,1,2)    38: (3,1,2)        69: (4,2,1)
    16: (5)        40: (2,4)          70: (4,1,2)
    19: (3,1,1)    42: (2,2,2)        72: (3,4)
    20: (2,3)      44: (2,1,3)        73: (3,3,1)
    21: (2,2,1)    47: (2,1,1,1,1)    74: (3,2,2)
    22: (2,1,2)    48: (1,5)          76: (3,1,3)
    24: (1,4)      51: (1,3,1,1)      79: (3,1,1,1,1)

The version for prime indices is A000037.
The version for Heinz numbers of partitions is A026424, counted by A027193.
These compositions are counted by A027306.
These are the positions of terms > 0 in A344618.
The weak (k >= 0) version is A345914.
The version for unreversed alternating sum is A345917.
The opposite (k < 0) version is A345920.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000290, A000346, A008549, A025047, A027187, A028260, A034871, A114121, A163493, A344607, A344608, A344650, A344743, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]>0&]

A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

Original entry on oeis.org

1, 4, 6, 15, 16, 21, 24, 25, 35, 60, 64, 77, 84, 90, 91, 96, 100, 121, 126, 140, 143, 150, 210, 221, 240, 247, 256, 289, 297, 308, 323, 336, 351, 360, 364, 375, 384, 400, 437, 462, 484, 490, 495, 504, 525, 529, 546, 551, 560, 572, 585, 600, 625, 667, 686, 726

Offset: 1

Gus Wiseman, Jan 21 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers with the same number of even prime indices as odd conjugate prime indices.

These are also partitions for which the number of even parts is equal to the positive alternating sum of the parts.

			The terms and their prime indices begin:
    1: ()
    4: (1,1)
    6: (2,1)
   15: (3,2)
   16: (1,1,1,1)
   21: (4,2)
   24: (2,1,1,1)
   25: (3,3)
   35: (4,3)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
   77: (5,4)
   84: (4,2,1,1)
   90: (3,2,2,1)
   91: (6,4)
   96: (2,1,1,1,1,1)

A subset of A028260 (even bigomega), counted by A027187.
These partitions are counted by A277579.
This is the half-conjugate version of A325698, counted by A045931.
A000041 counts partitions, strict A000009.
A047993 counts balanced partitions, ranked by A106529.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A100824 counts partitions with at most one odd part, ranked by A349150.
A108950/A108949 count partitions with more odd/even parts.
A122111 represents conjugation using Heinz numbers.
A130780/A171966 count partitions with more or equal odd/even parts.
A257991/A257992 count odd/even prime indices.
A316524 gives the alternating sum of prime indices (reverse: A344616).
Cf. A000700, A000712, A035363, A066207, A066208, A097613, A215366, A239241, A240009, A241638, A316523, A325700, A340604.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(A122111(a(n))).

A345920 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum < 0.

Original entry on oeis.org

5, 9, 17, 18, 23, 25, 29, 33, 34, 39, 45, 49, 57, 65, 66, 68, 71, 75, 77, 78, 81, 85, 89, 90, 95, 97, 98, 103, 105, 109, 113, 114, 119, 121, 125, 129, 130, 132, 135, 139, 141, 142, 149, 153, 154, 159, 161, 169, 177, 178, 183, 189, 193, 194, 199, 205, 209, 217

Offset: 1

Gus Wiseman, Jul 09 2021

nonn

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

			The initial terms and the corresponding compositions:
      5: (2,1)         68: (4,3)
      9: (3,1)         71: (4,1,1,1)
     17: (4,1)         75: (3,2,1,1)
     18: (3,2)         77: (3,1,2,1)
     23: (2,1,1,1)     78: (3,1,1,2)
     25: (1,3,1)       81: (2,4,1)
     29: (1,1,2,1)     85: (2,2,2,1)
     33: (5,1)         89: (2,1,3,1)
     34: (4,2)         90: (2,1,2,2)
     39: (3,1,1,1)     95: (2,1,1,1,1,1)
     45: (2,1,2,1)     97: (1,5,1)
     49: (1,4,1)       98: (1,4,2)
     57: (1,1,3,1)    103: (1,3,1,1,1)
     65: (6,1)        105: (1,2,3,1)
     66: (5,2)        109: (1,2,1,2,1)

The version for prime indices is {}.
The version for Heinz numbers of partitions is A119899.
These compositions are counted by A294175  (even bisection: A008549).
These are the positions of terms < 0 in A344618.
The complement is A345914.
The weak (k <= 0) version is A345916.
The opposite (k > 0) version is A345918.
The version for unreversed alternating sum is A345919.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]<0&]

A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 102, 342, 1213, 4361, 16016, 59348, 222117, 836315, 3166852, 12042620, 45967479, 176005709, 675759564, 2600672458, 10029832754, 38753710486, 149990133774, 581393603996, 2256710139346, 8770547818956, 34125389919850, 132919443189544, 518232001761434, 2022337118015338, 7898574056034636, 30873421455729728

Offset: 0

N. J. A. Sloane

In other words, a(n) is the number of graphic sequences of length n, where a graphic sequence is a sequence of numbers which can be the degree sequence of some graph.

In the article by A. Iványi, G. Gombos, L. Lucz, and T. Matuszka, "Parallel enumeration of degree sequences of simple graphs II", in Table 4 on page 260 the values a(30) = 7898574056034638 and a(31) = 30873429530206738 are incorrect due to the incorrect Gz(30) = 5876236938019300 and Gz(31) = 22974847474172100. - Wang Kai, Jun 05 2016

			For n = 3, there are 4 different graphic sequences possible: 0 0 0; 1 1 0; 2 1 1; 2 2 2. - Daan van Berkel (daan.v.berkel.1980(AT)gmail.com), Jun 25 2010
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(0) = 1 through a(4) = 11 sorted degree sequences:
  ()  (0)  (0,0)  (0,0,0)  (0,0,0,0)
           (1,1)  (0,1,1)  (0,0,1,1)
                  (1,1,2)  (0,1,1,2)
                  (2,2,2)  (0,2,2,2)
                           (1,1,1,1)
                           (1,1,1,3)
                           (1,1,2,2)
                           (1,2,2,3)
                           (2,2,2,2)
                           (2,2,3,3)
                           (3,3,3,3)
For example, the graph {{2,3},{2,4}} has degrees (0,2,1,1), so (0,1,1,2) is counted under a(4).
(End)

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).

Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Table of n, a(n) for n = 0..1651 (Terms 1 through 31 were computed by various authors; terms 32 through 34 by Axel Kohnert and Wang Kai; terms 35 to 79 by Wang Kai)
Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Counting graphic sequences via integrated random walks, arXiv:2301.07022 [math.CO], 2023.
T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995), #R11.
B. A. Chat, S. Pirzada, and A. Iványi, Recognition of split-graphic sequences, Acta Universitatis Sapientiae, Informatica, 6, 2 (2014) 252-286.
D. Dimitrov, Efficient computation of trees with minimal atom-bond connectivity index, arXiv:1305.1155 [cs.DM], 2013.
A. Iványi, L. Lucz, T. F. Móri and P. Sótér, On Erdős-Gallai and Havel-Hakimi algorithms, Acta Univ. Sapientiae, Inform. 3(2) (2011), 230-268.
A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012), 260-288. - From _N. J. A. Sloane_, Feb 15 2013
A. Ivanyi and J. E. Schoenfield, Deciding football sequences, Acta Univ. Sapientiae, Informatica, 4, 1 (2012), 130-183. - From _N. J. A. Sloane_, Dec 22 2012 [Disclaimer: I am not one of the authors of this paper; I was unpleasantly surprised to find my name on it, as explained here. - _Jon E. Schoenfield_, Nov 26 2016]
Wang Kai, Efficient Counting of Degree Sequences, arXiv:1604.04148 [math.CO], 2016. Gives 79 terms.
P. W. Mills, R. P. Rundle, V. M. Dwyer, T. Tilma, and S. J. Devitt, A proposal for an efficient quantum algorithm solving the graph isomorphism problem, arXiv 1711.09842, 2017.
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Degree Sequence.
Eric Weisstein's World of Mathematics, Graphic Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.
Index entries for sequences related to graphical partitions

Counting the positive partitions by sum gives A000569, ranked by A320922.
The version with half-loops is A029889, with covering case A339843.
The covering case (no zeros) is A095268.
Covering simple graphs are ranked by A309356 and A320458.
Non-graphical partitions are counted by A339617 and ranked by A339618.
The version with loops is A339844, with covering case A339845.
A006125 counts simple graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320921 counts connected graphical partitions.
A322353 counts factorizations into distinct semiprimes.
A339659 counts graphical partitions of 2n into k parts.
A339661 counts factorizations into distinct squarefree semiprimes.
Cf. A004250, A007717, A181819, A320894, A320923, A339560, A339561, A339842.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{2}]]]],{n,0,5}] (* Gus Wiseman, Dec 31 2020 *)

G.f. = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 102*x^6 + 342*x^7 + 1213*x^8 + ...

a(n) ~ c * 4^n / n^(3/4) for some constant c > 0. Computational estimates suggest c ≈ 0.099094. - Tom Johnston, Jan 18 2023

More terms from Torsten Sillke, torsten.sillke(AT)lhsystems.com, using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.
a(19) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 19 2007
a(20)-a(23) from Nathann Cohen, Jul 09 2011
a(24)-a(29) from Antal Iványi, Nov 15 2011
a(30) and a(31) corrected by Wang Kai, Jun 05 2016

A174725 a(n) = (A074206(n) + A008683(n))/2.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2

Offset: 1

Mats Granvik, Mar 28 2010

nonn

From Mats Granvik, May 25 2017: (Start)
A074206(n) = A002033(n-1) = a(n) + A174726(n).
A008683(n) = a(n) - A174726(n).
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
  (2*6)  (3*8)      (5*6)   (4*8)      (4*9)
  (3*4)  (4*6)      (6*5)   (8*4)      (6*6)
  (4*3)  (6*4)      (10*3)  (16*2)     (9*4)
  (6*2)  (8*3)      (15*2)  (2*16)     (12*3)
         (12*2)     (2*15)  (2*2*2*4)  (18*2)
         (2*12)     (3*10)  (2*2*4*2)  (2*18)
         (2*2*2*3)          (2*4*2*2)  (3*12)
         (2*2*3*2)          (4*2*2*2)  (2*2*3*3)
         (2*3*2*2)                     (2*3*2*3)
         (3*2*2*2)                     (2*3*3*2)
                                       (3*2*2*3)
                                       (3*2*3*2)
                                       (3*3*2*2)
(End)

Mats Granvik, Table of n, a(n) for n = 1..10000

Cf. A008683, A051731.
The odd version is A174726.
The unordered version is A339846.
A001055 counts factorizations, with strict case A045778.
A058696 counts partitions of even numbers, ranked by A300061.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
Cf. A002033, A027193, A028260, A050320, A058695, A236913, A316439, A320655, A320656, A339890, A378216 (Dirichlet inverse).

(* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],EvenQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A074206.
a(n) = (A074206(n) + A008683(n))/2.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019

References to A002033(n-1) changed to A074206(n) by Antti Karttunen, Nov 23 2024

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

Original entry on oeis.org

1, 6, 8, 14, 15, 20, 26, 27, 33, 35, 36, 38, 44, 48, 50, 51, 58, 63, 64, 65, 68, 69, 74, 77, 84, 86, 90, 92, 93, 95, 106, 110, 112, 117, 119, 120, 122, 123, 124, 125, 141, 142, 143, 145, 147, 156, 158, 160, 161, 162, 164, 170, 171, 177, 178, 185, 188, 196, 198, 201, 202, 208, 209, 210, 214, 215, 216, 217, 219, 221, 225

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.
The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023

Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
Subsequence of {1} U A268388.

Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332820(n) =  { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };

{a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
{a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
From Peter Munn, Mar 17 2021: (Start)
{a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
{a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
{a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
{a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
(End)

New name from Peter Munn, Mar 08 2021

A332823 A 3-way classification indicator generated by the products of two consecutive primes and the cubes of primes. a(n) is -1, 0, or 1 such that a(n) == A048675(n) (mod 3).

Original entry on oeis.org

0, 1, -1, -1, 1, 0, -1, 0, 1, -1, 1, 1, -1, 0, 0, 1, 1, -1, -1, 0, 1, -1, 1, -1, -1, 0, 0, 1, -1, 1, 1, -1, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, -1, 0, -1, -1, 1, 0, 1, 0, 0, 1, -1, 1, -1, -1, 1, 0, 1, -1, -1, -1, 0, 0, 0, 1, 1, 0, 0, 1, -1, 1, 1, 0, 1, 1, 0, -1, -1, -1, -1, -1, 1, 0, -1, 0, 1, 1, -1, 0

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

sign

Completely additive modulo 3.
The equivalent sequence modulo 2 is A096268 (with offset 1), which produces the {A003159, A036554} classification.
Let H be the multiplicative subgroup of the positive rational numbers generated by the products of two consecutive primes and the cubes of primes. a(n) indicates the coset of H containing n. a(n) = 0 if n is in H. a(n) = 1 if n is in 2H. a(n) = -1 if n is in (1/2)H.
The properties of this classification can usefully be compared to two well-studied classifications. With the {A026424, A028260} classes, multiplying a member of one class by a prime gives a member of the other class. With the {A000028, A000379} classes, adding a factor to the Fermi-Dirac factorization of a member of one class gives a member of the other class. So, if 4 is not a Fermi-Dirac factor of k, k and 4k will be in different classes of the {A000028, A000379} set; but k and 4k will be in the same class of the {A026424, A028260} set. For two numbers to necessarily be in different classes when they differ in either of the 2 ways described above, 3 classes are needed.
With the classes defined by this sequence, no two of k, 2k and 4k are in the same class. This is a consequence of the following stronger property: if k is any positive integer and m is a member of A050376 (often called Fermi-Dirac primes), then no two of k, k * m, k * m^2 are in the same class. Also, if p and q are consecutive primes, then k * p and k * q are in different classes.
Further properties are given in the sequences that list the classes: A332820, A332821, A332822.
The scaled imaginary part of the Eisenstein integer-valued function, f, defined in A353445. - Peter Munn, Apr 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A332813 (0,1,2 version of this sequence), A353350.
Cf. A353354 (inverse Möbius transform, gives another 3-way classification indicator function).
Cf. A102283, A048675.
Cf. A332820, A332821, A332822 for positions of 0's, 1's and -1's in this sequence; also A003159, A036554 for the modulo 2 equivalents.
Comparable functions: A008836, A064179, A096268, A332814.
A000035, A003961, A028234, A055396, A067029, A097248, A225546, A297845, A331590 are used to express relationship between terms of this sequence.
The formula section also details how the sequence maps the terms of A000040, A332461, A332462.

A332823(n) = { my(f = factor(n),u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u,-1,u); };

a(n) = A102283(A048675(n)) = -1 + (1 + A048675(n)) mod 3.
a(1) = 0; for n > 1, a(n) = A102283[(A067029(n) * (2-(A000035(A055396(n))))) + a(A028234(n))].
For all n >= 1, k >= 1: (Start)
a(n * k) == a(n) + a(k) (mod 3).
a(A331590(n,k)) == a(n) + a(k) (mod 3).
a(n^2) = -a(n).
a(A003961(n)) = -a(n).
a(A297845(n,k)) = a(n) * a(k).
(End)
For all n >= 1: (Start)
a(A000040(n)) = (-1)^(n-1).
a(A225546(n)) = a(n).
a(A097248(n)) = a(n).
a(A332461(n)) = a(A332462(n)) = A332814(n).
(End)
a(n) = A332814(A332462(n)). [Compare to the formula above. For a proof, see A353350.] - Antti Karttunen, Apr 16 2022

cp's OEIS Frontend

A340101 Number of factorizations of 2n + 1 into odd factors > 1.

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A345919 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum < 0.

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A340601 Number of integer partitions of n of even rank.

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A345918 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum > 0.

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A349157 Heinz numbers of integer partitions where the number of even parts is equal to the number of odd conjugate parts.

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A345920 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum < 0.

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A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

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A174725 a(n) = (A074206(n) + A008683(n))/2.

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