A357631 -id:A357631 - cp's OEIS frontend

A357706 Numbers k such that the k-th composition in standard order has half-alternating sum and skew-alternating sum both 0.

0, 15, 45, 54, 59, 153, 170, 179, 204, 213, 230, 235, 247, 255, 561, 594, 611, 660, 677, 710, 715, 727, 735, 750, 765, 792, 809, 842, 851, 871, 879, 894, 908, 917, 934, 939, 951, 959, 973, 982, 987, 1005, 1014, 1019

Offset: 1

Gus Wiseman, Oct 13 2022

nonn

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ..., and the skew-alternating sum to be A - B - C + D + E - F - G + ...

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.

For partitions and half only (or both): A000583, counted by A035363.
These compositions are counted by A228248.
For half-alternating only: A357625, reverse A357626, counted by A357641.
For skew-alternating only: A357627, reverse A357628, counted by A001700.
For reversed partitions and half only: A357631, counted by A357639.
For reversed partitions and skew only A357632, counted by A357640.
For partitions and skew only: A357636, counted by A035594.
Cf. A001511, A004006, A088218, A357136, A357182, A357642.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Select[Range[0,1000],halfats[stc[#]]==0&&skats[stc[#]]==0&]

Intersection of A357625 and A357627.

A357851 Numbers k such that the half-alternating sum of the prime indices of k is 1.

Original entry on oeis.org

2, 8, 18, 32, 45, 50, 72, 98, 105, 128, 162, 180, 200, 231, 242, 275, 288, 338, 392, 420, 429, 450, 455, 512, 578, 648, 663, 720, 722, 800, 833, 882, 924, 935, 968, 969, 1050, 1058, 1100, 1125, 1152, 1235, 1250, 1311, 1352, 1458, 1463, 1568, 1680, 1682, 1716

Offset: 1

Gus Wiseman, Oct 28 2022

nonn

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

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 terms together with their prime indices begin:
     2: {1}
     8: {1,1,1}
    18: {1,2,2}
    32: {1,1,1,1,1}
    45: {2,2,3}
    50: {1,3,3}
    72: {1,1,1,2,2}
    98: {1,4,4}
   105: {2,3,4}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   180: {1,1,2,2,3}
   200: {1,1,1,3,3}

The version for k = 0 is A357631, standard compositions A357625-A357626.
The version for original alternating sum is A001105.
Positions of ones in A357629, reverse A357633.
The skew version for k = 0 is A357632, reverse A357636.
Partitions with these Heinz numbers are counted by A035444, skew A035544.
The reverse version is A357635, k = 0 version A000583.
A056239 adds up prime indices, row sums of A112798.
A316524 gives alternating sum of prime indices, reverse A344616.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357641 counts comps w/ half-alt sum 0, even-length A357642.
Cf. A003963, A053251, A055932, A345958, A357621-A357624, A357630, A357634, A357637, A357639, A357640.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
Select[Range[1000],halfats[primeMS[#]]==1&]

cp's OEIS Frontend

A357706 Numbers k such that the k-th composition in standard order has half-alternating sum and skew-alternating sum both 0.

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