A344616 -id:A344616 - cp's OEIS frontend

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

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 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:
     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&]

A345958 Numbers whose prime indices have reverse-alternating sum 1.

Original entry on oeis.org

2, 6, 8, 15, 18, 24, 32, 35, 50, 54, 60, 72, 77, 96, 98, 128, 135, 140, 143, 150, 162, 200, 216, 221, 240, 242, 288, 294, 308, 315, 323, 338, 375, 384, 392, 437, 450, 486, 512, 540, 560, 572, 578, 600, 648, 667, 693, 722, 726, 735, 800, 864, 875, 882, 884, 899

Offset: 1

Gus Wiseman, Jul 11 2021

nonn

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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
Also numbers with exactly one odd conjugate prime index. Conjugate prime indices are listed by A321650, ranked by A122111.

			The initial terms and their prime indices:
   2: {1}
   6: {1,2}
   8: {1,1,1}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  35: {3,4}
  50: {1,3,3}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  77: {4,5}
  96: {1,1,1,1,1,2}
  98: {1,4,4}

The k > 0 version is A000037.
These multisets are counted by A000070.
The k = 0 version is A000290, counted by A000041.
The version for unreversed-alternating sum is A001105.
These partitions are counted by A035363.
These are the positions of 1's in A344616.
The k = 2 version is A345961, counted by A120452.
A000984/A345909/A345911 count/rank compositions with alternating sum 1.
A001791/A345910/A345912 count/rank compositions with alternating sum -1.
A088218 counts compositions with alternating sum 0, ranked by A344619.
A025047 counts wiggly compositions.
A027187 counts partitions with reverse-alternating sum <= 0.
A056239 adds up prime indices, row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices.
A325534 and A325535 count separable and inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344607 counts partitions with reverse-alternating sum >= 0.
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000097, A027193, A034871, A239830, A341446, A344650, A344651, A344743, A345917, A345918, A345920.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[100],sats[primeMS[#]]==1&]

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 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:
      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&]

A345921 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 44, 45, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jul 10 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.
Also numbers k such that the k-th composition in standard order has reverse-alternating sum != 0.

			The initial terms and the corresponding compositions:
     1: (1)        20: (2,3)          35: (4,1,1)
     2: (2)        21: (2,2,1)        37: (3,2,1)
     4: (3)        22: (2,1,2)        38: (3,1,2)
     5: (2,1)      23: (2,1,1,1)      39: (3,1,1,1)
     6: (1,2)      24: (1,4)          40: (2,4)
     7: (1,1,1)    25: (1,3,1)        42: (2,2,2)
     8: (4)        26: (1,2,2)        44: (2,1,3)
     9: (3,1)      27: (1,2,1,1)      45: (2,1,2,1)
    11: (2,1,1)    28: (1,1,3)        47: (2,1,1,1,1)
    12: (1,3)      29: (1,1,2,1)      48: (1,5)
    14: (1,1,2)    30: (1,1,1,2)      49: (1,4,1)
    16: (5)        31: (1,1,1,1,1)    51: (1,3,1,1)
    17: (4,1)      32: (6)            52: (1,2,3)
    18: (3,2)      33: (5,1)          54: (1,2,1,2)
    19: (3,1,1)    34: (4,2)          56: (1,1,4)

The version for Heinz numbers of partitions is A000037.
These compositions are counted by A058622.
These are the positions of terms != 0 in A124754.
The complement (k = 0) is A344619.
The positive (k > 0) version is A345917 (reverse: A345918).
The negative (k < 0) version is A345919 (reverse: 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).
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, A008549, A025047, A032443, A034871, A114121, A163493, A236913, A344609, A344651, 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&]

A345914 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

0, 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 67, 69, 70, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 87, 88

Offset: 1

Gus Wiseman, Jul 04 2021

nonn

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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 sequence of terms together with the corresponding compositions begins:
     0: ()           19: (3,1,1)        40: (2,4)
     1: (1)          20: (2,3)          41: (2,3,1)
     2: (2)          21: (2,2,1)        42: (2,2,2)
     3: (1,1)        22: (2,1,2)        43: (2,2,1,1)
     4: (3)          24: (1,4)          44: (2,1,3)
     6: (1,2)        26: (1,2,2)        46: (2,1,1,2)
     7: (1,1,1)      27: (1,2,1,1)      47: (2,1,1,1,1)
     8: (4)          28: (1,1,3)        48: (1,5)
    10: (2,2)        30: (1,1,1,2)      50: (1,3,2)
    11: (2,1,1)      31: (1,1,1,1,1)    51: (1,3,1,1)
    12: (1,3)        32: (6)            52: (1,2,3)
    13: (1,2,1)      35: (4,1,1)        53: (1,2,2,1)
    14: (1,1,2)      36: (3,3)          54: (1,2,1,2)
    15: (1,1,1,1)    37: (3,2,1)        55: (1,2,1,1,1)
    16: (5)          38: (3,1,2)        56: (1,1,4)

The version for prime indices is A000027, counted by A000041.
These compositions are counted by A116406.
The case of non-Heinz numbers of partitions is A119899, counted by A344608.
The version for Heinz numbers of partitions is A344609, counted by A344607.
These are the positions of terms >= 0 in A344618.
The version for unreversed alternating sum is A345913.
The opposite (k <= 0) version is A345916.
The strict (k > 0) case is A345918.
The complement is A345920, counted by A294175.
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, A000346, A008549, A025047, A027187, A032443, A034871, A114121, A120452, A163493, A238279, A344650, A344743.

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&]

A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.

Original entry on oeis.org

0, 3, 6, 10, 12, 13, 15, 20, 24, 25, 27, 30, 36, 40, 41, 43, 46, 48, 49, 50, 51, 53, 54, 55, 58, 60, 61, 63, 72, 80, 81, 83, 86, 92, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 111, 116, 120, 121, 123, 126, 136, 144, 145, 147, 150, 156, 160, 161, 162, 163

Offset: 1

Gus Wiseman, Jul 08 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 sequence of terms together with the corresponding compositions begins:
     0: ()
     3: (1,1)
     6: (1,2)
    10: (2,2)
    12: (1,3)
    13: (1,2,1)
    15: (1,1,1,1)
    20: (2,3)
    24: (1,4)
    25: (1,3,1)
    27: (1,2,1,1)
    30: (1,1,1,2)
    36: (3,3)
    40: (2,4)
    41: (2,3,1)

The version for Heinz numbers of partitions is A028260 (counted by A027187).
These compositions are counted by A058622.
These are the positions of terms <= 0 in A124754.
The reverse-alternating version is A345916.
The opposite (k >= 0) version is A345917.
The strictly negative (k < 0) version 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, A000097, A000346, A008549, A025047, A032443, A114121, A163493, A344607, A344609, A344610.

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&]

A345916 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

0, 3, 5, 9, 10, 13, 15, 17, 18, 23, 25, 29, 33, 34, 36, 39, 41, 43, 45, 46, 49, 50, 53, 55, 57, 58, 61, 63, 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, 136, 139, 141, 142, 145, 147, 149

Offset: 1

Gus Wiseman, Jul 08 2021

nonn

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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 sequence of terms together with the corresponding compositions begins:
     0: ()
     3: (1,1)
     5: (2,1)
     9: (3,1)
    10: (2,2)
    13: (1,2,1)
    15: (1,1,1,1)
    17: (4,1)
    18: (3,2)
    23: (2,1,1,1)
    25: (1,3,1)
    29: (1,1,2,1)
    33: (5,1)
    34: (4,2)
    36: (3,3)

The version for Heinz numbers of partitions is A000290.
These compositions are counted by A058622.
These are the positions of terms <= 0 in A344618.
The opposite (k >= 0) version is A345914.
The version for unreversed alternating sum is A345915.
The strictly negative (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).
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, A000346, A008549, A025047, A027187, A028260, A032443, A114121, A163493, A344607, A344610, 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&]

A344650 Number of strict odd-length integer partitions of 2n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 11, 16, 23, 32, 44, 61, 82, 111, 148, 195, 256, 334, 432, 557, 713, 908, 1152, 1455, 1829, 2291, 2859, 3554, 4404, 5440, 6697, 8222, 10066, 12288, 14964, 18176, 22023, 26625, 32117, 38656, 46432, 55661, 66592, 79523, 94793, 112792, 133984

Offset: 0

Gus Wiseman, Jun 05 2021

nonn

Also the number of strict integer partitions of 2n with reverse-alternating sum >= 0.

Also the number of reversed strict integer partitions of 2n with alternating sum >= 0.

			The a(1) = 1 through a(8) = 16 partitions:
  (2)  (4)  (6)      (8)      (10)     (12)     (14)      (16)
            (3,2,1)  (4,3,1)  (5,3,2)  (5,4,3)  (6,5,3)   (7,5,4)
                     (5,2,1)  (5,4,1)  (6,4,2)  (7,4,3)   (7,6,3)
                              (6,3,1)  (6,5,1)  (7,5,2)   (8,5,3)
                              (7,2,1)  (7,3,2)  (7,6,1)   (8,6,2)
                                       (7,4,1)  (8,4,2)   (8,7,1)
                                       (8,3,1)  (8,5,1)   (9,4,3)
                                       (9,2,1)  (9,3,2)   (9,5,2)
                                                (9,4,1)   (9,6,1)
                                                (10,3,1)  (10,4,2)
                                                (11,2,1)  (10,5,1)
                                                          (11,3,2)
                                                          (11,4,1)
                                                          (12,3,1)
                                                          (13,2,1)
                                                          (6,4,3,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

The Heinz numbers are the intersection of A030059 and A300061.
Allowing even length gives A035294 (non-strict: A058696).
Even bisection of A067659.
The opposite type of strict partition (even length and odd sum) is A343942.
The non-strict version is A236559 or A344611.
Row sums of A344649.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A124754 gives alternating sums of standard compositions (reverse: A344618).
A152146 interleaved with A152157 counts strict partitions by sum and alternating sum.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A343941 counts strict partitions of 2n with reverse-alternating sum 4.
A344604 counts wiggly compositions with twins.
A344739 counts strict partitions by sum and reverse-alternating sum.
A344741 counts partitions of 2n with reverse-alternating sum -2.
Cf. A000070, A000097, A027187, A114121, A239829, A239830, A344607, A344609, A344610, A344651, A344654.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30,2}]

Sum of odd-indexed terms in row 2n of A008289.

a(n) = A067659(2n).

A344742 Numbers whose prime factors have a permutation with no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Jun 12 2021

nonn

Differs from A335433 in having all squares of primes (A001248) and lacking 270 etc.
Also Heinz numbers of integer partitions that are either a twin (x,x) or have a wiggly permutation.
(1) The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
(2) A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The sequence of terms together with their prime indices begins:
      1: {}          18: {1,2,2}     36: {1,1,2,2}
      2: {1}         19: {8}         37: {12}
      3: {2}         20: {1,1,3}     38: {1,8}
      4: {1,1}       21: {2,4}       39: {2,6}
      5: {3}         22: {1,5}       41: {13}
      6: {1,2}       23: {9}         42: {1,2,4}
      7: {4}         25: {3,3}       43: {14}
      9: {2,2}       26: {1,6}       44: {1,1,5}
     10: {1,3}       28: {1,1,4}     45: {2,2,3}
     11: {5}         29: {10}        46: {1,9}
     12: {1,1,2}     30: {1,2,3}     47: {15}
     13: {6}         31: {11}        49: {4,4}
     14: {1,4}       33: {2,5}       50: {1,3,3}
     15: {2,3}       34: {1,7}       51: {2,7}
     17: {7}         35: {3,4}       52: {1,1,6}
For example, the prime factors of 120 are (2,2,2,3,5), with the two wiggly permutations (2,3,2,5,2) and (2,5,2,3,2), so 120 is in the sequence.

Positions of nonzero terms in A344606.
The complement is A344653, counted by A344654.
These partitions are counted by A344740.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A001248 lists squares of primes.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A011782 counts compositions.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A056239 adds up prime indices,  row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts wiggly compositions with twins.
A345164 counts wiggly permutations of prime indices.
A345165 counts partitions without a wiggly permutation, ranked by A345171.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A000070, A001222, A071321, A071322, A316523, A316524, A344605, A344614, A344616, A344652, A345163, A345166, A345167, A345173.

Select[Range[100],Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]!={}&]

Union of A345172 (wiggly) and A001248 (squares of primes).

cp's OEIS Frontend

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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A345958 Numbers whose prime indices have reverse-alternating sum 1.

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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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A345921 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum != 0.

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A345914 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum >= 0.

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A345915 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum <= 0.

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A345916 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum <= 0.

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A344650 Number of strict odd-length integer partitions of 2n.

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