A000097 -id:A000097 - cp's OEIS frontend

A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even.

6, 10, 14, 15, 21, 22, 24, 26, 33, 34, 35, 38, 39, 40, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 104, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 135, 136, 140, 141, 142, 143, 145, 146, 150

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Also numbers whose alternating sum of prime indices is < 0. Equivalently, numbers with even bigomega whose conjugate prime indices are not all even. This is the intersection of A028260 and A000037. - Gus Wiseman, Jun 20 2021

			From _Gus Wiseman_, Jun 20 2021: (Start)
The sequence of terms together with their prime indices begins:
       6: {1,2}          51: {2,7}          86: {1,14}
      10: {1,3}          54: {1,2,2,2}      87: {2,10}
      14: {1,4}          55: {3,5}          88: {1,1,1,5}
      15: {2,3}          56: {1,1,1,4}      90: {1,2,2,3}
      21: {2,4}          57: {2,8}          91: {4,6}
      22: {1,5}          58: {1,10}         93: {2,11}
      24: {1,1,1,2}      60: {1,1,2,3}      94: {1,15}
      26: {1,6}          62: {1,11}         95: {3,8}
      33: {2,5}          65: {3,6}          96: {1,1,1,1,1,2}
      34: {1,7}          69: {2,9}         104: {1,1,1,6}
      35: {3,4}          74: {1,12}        106: {1,16}
      38: {1,8}          77: {4,5}         111: {2,12}
      39: {2,6}          82: {1,13}        115: {3,9}
      40: {1,1,1,3}      84: {1,1,2,4}     118: {1,17}
      46: {1,9}          85: {3,7}         119: {4,7}
(End)

Superset: A119847. Subset: A006881. The intersection of A028260 and A000037.
Positions of negative terms in A316524.
The partitions with these Heinz numbers are counted by A344608.
Complement of A344609.
Cf. A000041, A000070, A000097, A027187, A103919, A116406, A239829, A239830, A343941, A344607, A344651.

Select[Range[200],And@@EvenQ[{PrimeOmega[#],DivisorSigma[0,#]}]&] (* Harvey P. Dale, Jan 24 2013 *)

A344609 Numbers whose alternating sum of prime indices is >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 30, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 102, 103, 105, 107

Offset: 1

Gus Wiseman, May 30 2021

nonn

Also Heinz numbers of partitions whose reverse-alternating sum is >= 0. These are partitions whose conjugate parts are all even or whose length is odd.
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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			The sequence of terms together with their prime indices begins:
      1: {}            20: {1,1,3}         45: {2,2,3}
      2: {1}           23: {9}             47: {15}
      3: {2}           25: {3,3}           48: {1,1,1,1,2}
      4: {1,1}         27: {2,2,2}         49: {4,4}
      5: {3}           28: {1,1,4}         50: {1,3,3}
      7: {4}           29: {10}            52: {1,1,6}
      8: {1,1,1}       30: {1,2,3}         53: {16}
      9: {2,2}         31: {11}            59: {17}
     11: {5}           32: {1,1,1,1,1}     61: {18}
     12: {1,1,2}       36: {1,1,2,2}       63: {2,2,4}
     13: {6}           37: {12}            64: {1,1,1,1,1,1}
     16: {1,1,1,1}     41: {13}            66: {1,2,5}
     17: {7}           42: {1,2,4}         67: {19}
     18: {1,2,2}       43: {14}            68: {1,1,7}
     19: {8}           44: {1,1,5}         70: {1,3,4}
For example, the prime indices of 70 are {1,3,4} with alternating sum 1 - 3 + 4 = 2, so 70 is in the sequence. On the other hand, the prime indices of 24 are {1,1,1,2} with alternating sum 1 - 1 + 1 - 2 = -1, so 24 is not in the sequence.

The opposite (nonpositive) version is A028260, counted by A027187.
The strict case (n > 0) is counted by A067659, odd bisection A344650.
Permutations of prime indices of these terms are counted by A116406.
Complement of A119899, Heinz numbers of the partitions counted by A344608.
Positions of nonnegative terms in A316524 or A344617.
Heinz numbers of the partitions counted by A344607.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A000097 counts partitions with alternating sum 2.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum.
A120452 counts partitions with reverse-alternating sum 2.
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344612 counts partitions by sum and reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
Cf. A001222, A001250, A003242, A005649, A026424, A071321/A071322, A124754, A239829, A343938, A344611, A344651, A344653/A344742, A344739.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],ats[primeMS[#]]>=0&]

A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.

Original entry on oeis.org

6, 20, 25, 27, 30, 72, 81, 83, 86, 92, 98, 101, 103, 106, 109, 111, 116, 121, 123, 126, 272, 289, 291, 294, 300, 312, 322, 325, 327, 330, 333, 335, 340, 345, 347, 350, 360, 369, 371, 374, 380, 388, 393, 395, 398, 402, 405, 407, 410, 413, 415, 420, 425, 427

Offset: 1

Gus Wiseman, Jul 01 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:
      6: (1,2)
     20: (2,3)
     25: (1,3,1)
     27: (1,2,1,1)
     30: (1,1,1,2)
     72: (3,4)
     81: (2,4,1)
     83: (2,3,1,1)
     86: (2,2,1,2)
     92: (2,1,1,3)
     98: (1,4,2)
    101: (1,3,2,1)
    103: (1,3,1,1,1)
    106: (1,2,2,2)
    109: (1,2,1,2,1)

These compositions are counted by A001791.
A version using runs of binary digits is A031444.
These are the positions of -1's in A124754.
The opposite (positive 1) version is A345909.
The reverse version is A345912.
The version for alternating sum of prime indices is A345959.
Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions of 2n+1 with alternating sum 1, ranked by A001105.
A011782 counts compositions.
A097805 counts compositions by sum and 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.
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. A000097, A000346, A008549, A025047, A027187, A031443, A031448, A114121, A119899, A126869, A238279, A344617.

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[#]]==-1&]

A345911 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 1.

Original entry on oeis.org

1, 6, 7, 20, 21, 26, 27, 30, 31, 72, 73, 82, 83, 86, 87, 92, 93, 100, 101, 106, 107, 110, 111, 116, 117, 122, 123, 126, 127, 272, 273, 290, 291, 294, 295, 300, 301, 312, 313, 324, 325, 330, 331, 334, 335, 340, 341, 346, 347, 350, 351, 360, 361, 370, 371, 374

Offset: 1

Gus Wiseman, Jul 01 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:
     1: (1)
     6: (1,2)
     7: (1,1,1)
    20: (2,3)
    21: (2,2,1)
    26: (1,2,2)
    27: (1,2,1,1)
    30: (1,1,1,2)
    31: (1,1,1,1,1)
    72: (3,4)
    73: (3,3,1)
    82: (2,3,2)
    83: (2,3,1,1)
    86: (2,2,1,2)
    87: (2,2,1,1,1)

These compositions are counted by A000984 (bisection of A126869).
The version for Heinz numbers of partitions is A001105.
A version using runs of binary digits is A066879.
These are positions of 1's in A344618.
The non-reverse version is A345909.
The opposite (negative 1) version is A345912.
The version for prime indices is A345958.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
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).
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.
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, A027193, A031448, A034871, A114121, A120452, A344607.

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[#]]==1&]

A345909 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 1.

Original entry on oeis.org

1, 5, 7, 18, 21, 23, 26, 29, 31, 68, 73, 75, 78, 82, 85, 87, 90, 93, 95, 100, 105, 107, 110, 114, 117, 119, 122, 125, 127, 264, 273, 275, 278, 284, 290, 293, 295, 298, 301, 303, 308, 313, 315, 318, 324, 329, 331, 334, 338, 341, 343, 346, 349, 351, 356, 361

Offset: 1

Gus Wiseman, Jun 30 2021

nonn

The alternating sum of a composition (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:
      1: (1)             87: (2,2,1,1,1)
      5: (2,1)           90: (2,1,2,2)
      7: (1,1,1)         93: (2,1,1,2,1)
     18: (3,2)           95: (2,1,1,1,1,1)
     21: (2,2,1)        100: (1,3,3)
     23: (2,1,1,1)      105: (1,2,3,1)
     26: (1,2,2)        107: (1,2,2,1,1)
     29: (1,1,2,1)      110: (1,2,1,1,2)
     31: (1,1,1,1,1)    114: (1,1,3,2)
     68: (4,3)          117: (1,1,2,2,1)
     73: (3,3,1)        119: (1,1,2,1,1,1)
     75: (3,2,1,1)      122: (1,1,1,2,2)
     78: (3,1,1,2)      125: (1,1,1,1,2,1)
     82: (2,3,2)        127: (1,1,1,1,1,1,1)
     85: (2,2,2,1)      264: (5,4)

These compositions are counted by A000984 (bisection of A126869).
The version for prime indices is A001105.
A version using runs of binary digits is A031448.
These are the positions of 1's in A124754.
The opposite (negative 1) version is A345910.
The reverse version is A345911.
The version for Heinz numbers of partitions is A345958.
Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
A000070 counts partitions with alternating sum 1 (ranked by A345957).
A000097 counts partitions with alternating sum 2 (ranked by A345960).
A011782 counts compositions.
A097805 counts compositions by sum and alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
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.
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 (this sequence)/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. A000346, A008549, A025047, A031443, A031444, A034871, A114121, A238279, A304620, A344651.

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[#]]==1&]

A345922 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 2.

Original entry on oeis.org

2, 11, 12, 14, 37, 40, 42, 47, 51, 52, 54, 59, 60, 62, 137, 144, 146, 151, 157, 163, 164, 166, 171, 172, 174, 181, 184, 186, 191, 197, 200, 202, 207, 211, 212, 214, 219, 220, 222, 229, 232, 234, 239, 243, 244, 246, 251, 252, 254, 529, 544, 546, 551, 557, 569

Offset: 1

Gus Wiseman, Jul 10 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:
      2: (2)            144: (3,5)
     11: (2,1,1)        146: (3,3,2)
     12: (1,3)          151: (3,2,1,1,1)
     14: (1,1,2)        157: (3,1,1,2,1)
     37: (3,2,1)        163: (2,4,1,1)
     40: (2,4)          164: (2,3,3)
     42: (2,2,2)        166: (2,3,1,2)
     47: (2,1,1,1,1)    171: (2,2,2,1,1)
     51: (1,3,1,1)      172: (2,2,1,3)
     52: (1,2,3)        174: (2,2,1,1,2)
     54: (1,2,1,2)      181: (2,1,2,2,1)
     59: (1,1,2,1,1)    184: (2,1,1,4)
     60: (1,1,1,3)      186: (2,1,1,2,2)
     62: (1,1,1,1,2)    191: (2,1,1,1,1,1,1)
    137: (4,3,1)        197: (1,4,2,1)

These compositions are counted by A088218.
The case of partitions is counted by A120452.
These are the positions of 2's in A344618.
The opposite (negative 2) version is A345923.
The version for unreversed alternating sum is A345925.
The version for Heinz numbers of partitions is A345961.
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).
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, A000097, A025047, A027193, A034871, A114121, A163493, A236913, A344607, A344608, A344741, 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[#]]==2&]

A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.

Original entry on oeis.org

9, 34, 39, 45, 49, 57, 132, 139, 142, 149, 154, 159, 161, 169, 178, 183, 189, 194, 199, 205, 209, 217, 226, 231, 237, 241, 249, 520, 531, 534, 540, 549, 554, 559, 564, 571, 574, 577, 585, 594, 599, 605, 612, 619, 622, 629, 634, 639, 642, 647, 653, 657, 665

Offset: 1

Gus Wiseman, Jul 10 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:
      9: (3,1)            183: (2,1,2,1,1,1)
     34: (4,2)            189: (2,1,1,1,2,1)
     39: (3,1,1,1)        194: (1,5,2)
     45: (2,1,2,1)        199: (1,4,1,1,1)
     49: (1,4,1)          205: (1,3,1,2,1)
     57: (1,1,3,1)        209: (1,2,4,1)
    132: (5,3)            217: (1,2,1,3,1)
    139: (4,2,1,1)        226: (1,1,4,2)
    142: (4,1,1,2)        231: (1,1,3,1,1,1)
    149: (3,2,2,1)        237: (1,1,2,1,2,1)
    154: (3,1,2,2)        241: (1,1,1,4,1)
    159: (3,1,1,1,1,1)    249: (1,1,1,1,3,1)
    161: (2,5,1)          520: (6,4)
    169: (2,2,3,1)        531: (5,3,1,1)
    178: (2,1,3,2)        534: (5,2,1,2)

These compositions are counted by A088218.
These are the positions of 2's in A344618.
The case of partitions of 2n is A344741.
The opposite (negative 2) version is A345923.
The version for unreversed alternating sum is A345925.
The version for Heinz numbers of partitions is A345961.
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).
A120452 counts partitions of 2n with reverse-alternating sum 2.
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, A000097, A000346, A025047, A032443, A034871, A106356, A163493, A236913, A344607, A344608, 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[#]]==-2&]

A345924 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -2.

Original entry on oeis.org

12, 40, 49, 51, 54, 60, 144, 161, 163, 166, 172, 184, 194, 197, 199, 202, 205, 207, 212, 217, 219, 222, 232, 241, 243, 246, 252, 544, 577, 579, 582, 588, 600, 624, 642, 645, 647, 650, 653, 655, 660, 665, 667, 670, 680, 689, 691, 694, 700, 720, 737, 739, 742

Offset: 1

Gus Wiseman, Jul 11 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:
     12: (1,3)          202: (1,3,2,2)        582: (3,4,1,2)
     40: (2,4)          205: (1,3,1,2,1)      588: (3,3,1,3)
     49: (1,4,1)        207: (1,3,1,1,1,1)    600: (3,2,1,4)
     51: (1,3,1,1)      212: (1,2,2,3)        624: (3,1,1,5)
     54: (1,2,1,2)      217: (1,2,1,3,1)      642: (2,6,2)
     60: (1,1,1,3)      219: (1,2,1,2,1,1)    645: (2,5,2,1)
    144: (3,5)          222: (1,2,1,1,1,2)    647: (2,5,1,1,1)
    161: (2,5,1)        232: (1,1,2,4)        650: (2,4,2,2)
    163: (2,4,1,1)      241: (1,1,1,4,1)      653: (2,4,1,2,1)
    166: (2,3,1,2)      243: (1,1,1,3,1,1)    655: (2,4,1,1,1,1)
    172: (2,2,1,3)      246: (1,1,1,2,1,2)    660: (2,3,2,3)
    184: (2,1,1,4)      252: (1,1,1,1,1,3)    665: (2,3,1,3,1)
    194: (1,5,2)        544: (4,6)            667: (2,3,1,2,1,1)
    197: (1,4,2,1)      577: (3,6,1)          670: (2,3,1,1,1,2)
    199: (1,4,1,1,1)    579: (3,5,1,1)        680: (2,2,2,4)

These compositions are counted by A002054.
These are the positions of -2's in A124754.
The version for reverse-alternating sum is A345923.
The opposite (positive 2) version is A345925.
The version for Heinz numbers of partitions is A345962.
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).
A120452 counts partitions of 2n with reverse-alternating sum 2.
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, A163493, A239830, A344609, A344651, A344741, A345908, A345961.

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[#]]==-2&]

A345925 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 2.

Original entry on oeis.org

2, 9, 11, 14, 34, 37, 39, 42, 45, 47, 52, 57, 59, 62, 132, 137, 139, 142, 146, 149, 151, 154, 157, 159, 164, 169, 171, 174, 178, 181, 183, 186, 189, 191, 200, 209, 211, 214, 220, 226, 229, 231, 234, 237, 239, 244, 249, 251, 254, 520, 529, 531, 534, 540, 546

Offset: 1

Gus Wiseman, Jul 11 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 corresponding compositions:
      2: (2)            137: (4,3,1)
      9: (3,1)          139: (4,2,1,1)
     11: (2,1,1)        142: (4,1,1,2)
     14: (1,1,2)        146: (3,3,2)
     34: (4,2)          149: (3,2,2,1)
     37: (3,2,1)        151: (3,2,1,1,1)
     39: (3,1,1,1)      154: (3,1,2,2)
     42: (2,2,2)        157: (3,1,1,2,1)
     45: (2,1,2,1)      159: (3,1,1,1,1,1)
     47: (2,1,1,1,1)    164: (2,3,3)
     52: (1,2,3)        169: (2,2,3,1)
     57: (1,1,3,1)      171: (2,2,2,1,1)
     59: (1,1,2,1,1)    174: (2,2,1,1,2)
     62: (1,1,1,1,2)    178: (2,1,3,2)
    132: (5,3)          181: (2,1,2,2,1)

These compositions are counted by A088218.
These are the positions of 2's in A124754.
The case of partitions of 2n is A344741.
The version for reverse-alternating sum is A345922.
The opposite (negative 2) version is A345924.
The version for Heinz numbers of partitions is A345960 (reverse: A345961).
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).
A120452 counts partitions of 2n with reverse-alternating sum 2.
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, A025047, A114121, A163493, A238279, A239830, A344607, A344608, A344609, A344651, A344743.

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[#]]==2&]

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&]

cp's OEIS Frontend

A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even.

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A344609 Numbers whose alternating sum of prime indices is >= 0.

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A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.

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A345911 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 1.

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A345909 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 1.

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A345922 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 2.

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A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.

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A345924 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -2.

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A345925 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 2.

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