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Sum_{d|n} mu(d)*a(n/d)^d = 1 for n > 0. - Andrew Howroyd, Feb 04 2023

0, 1, 2, 2, 5, 2, 12, 2, 26, 9, 36, 2, 206, 2, 132, 40, 677, 2, 1746, 2, 3398, 136, 2052, 2, 44388, 33, 8196, 730, 79166, 2, 263234, 2, 458330, 2056, 131076, 160, 8804349, 2, 524292, 8200, 13662156, 2, 36036674, 2, 48844526, 90282, 8388612, 2, 1971667502, 129

Andrew Howroyd, <a href="/A353860/b353860.txt">Table of n, a(n) for n = 0..1000</a>

(PARI) a(n) = if(n==0, 0, 1 - sumdiv(n, d, if(d>1, moebius(d)*a(n/d)^d ))) \\ Andrew Howroyd, Feb 04 2023

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Terms a(16) and beyond from Andrew Howroyd, Feb 04 2023

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A066099 lists compositions in standard order.

Cf. A237685, ~`A238279, A304442, `A318928, ~`A325277, `~A333381, A333755, A353844, `A353848, `A353849, A353850, `A353852.

repcams[q_List]:=repcams[q]=Union[{q}, If[UnsameQ@@q, {}, Union@@repcams/@ Union[Insert[Drop[q, #], Plus@@Take[q, #], First[#]]&/@ Select[Tuples[Range[Length[q]], 2], And[Less@@#, SameQ@@Take[q, #]]&]]]];

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], MemberQ[repcams[#], {n}]&]], {n, 0, 15}]

A003242 counts anti-run compositions , ranked by A333489, complement A261983.

A005811 counts runs in binary expansion.

`A318928 gives runs-resistance of binary expansion.

A351014 counts distinct runs in standard compositions, firsts A351015.

`A353840-A353846 pertain to partition run-sum trajectory.

Cf. A237685 ptns_dep_1, A238279 tri_comps_k_runs_wo_0, A304442 ptns_w_eq_runsums, A325277 red_topri_seq, A333381 stc_co_runs, A333755 tri_comps_runs, A353844 h_runsum_traj_last_prime, A353848 stc_w_eq_runsums, A353849 stc_numdstnct_runsums, A353850 comps_w_dstnct_runsums, A353852 stc_w_dsntct_runsums, A353855 stc_runsum_traj_last, A353858 comps_runsum_traj_last_sing.

Cf. A237685, ~`A238279, A304442, `A318928, ~`A325277, `~A333381, A333755, A353844, `A353848, `A353849, A353850, `A353852.

allocated for Gus WisemanNumber of collapsible integer compositions of n.

0, 1, 2, 2, 5, 2, 12, 2, 26, 9, 36, 2, 206, 2, 132, 40

0,3

If a collapse is a joining of some number of adjacent equal parts of an integer composition, we call a composition collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (1,1,1,3,2,1,1,2) -> (3,3,2,1,1,2) -> (3,3,2,2,2) -> (6,2,2,2) -> (6,6) -> (12), which shows that (1,1,1,3,2,1,1,2) is a collapsible composition of 12.

The a(0) = 0 through a(6) = 12 compositions:

. (1) (2) (3) (4) (5) (6)

(11) (111) (22) (11111) (33)

(112) (222)

(211) (1113)

(1111) (1122)

(2112)

(2211)

(3111)

(11112)

(11211)

(21111)

(111111)

repcams[q_List]:=repcams[q]=Union[{q}, If[UnsameQ@@q, {}, Union@@repcams/@Union[Insert[Drop[q, #], Plus@@Take[q, #], First[#]]&/@Select[Tuples[Range[Length[q]], 2], And[Less@@#, SameQ@@Take[q, #]]&]]]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[repcams[#], {n}]&]], {n, 0, 15}]

The version for partitions is A275870, ranked by A300273.

A003242 counts anti-run compositions ranked by A333489, complement A261983.

A005811 counts runs in binary expansion.

A011782 counts compositions.

A066099 lists compositions in standard order.

`A318928 gives runs-resistance of binary expansion.

A351014 counts distinct runs in standard compositions, firsts A351015.

`A353840-A353846 pertain to partition run-sum trajectory.

A353847 represents the run-sums of a composition, partitions A353832.

A353853-A353859 pertain to composition run-sum trajectory.

A353932 lists run-sums of standard compositions.

Cf. A237685 ptns_dep_1, A238279 tri_comps_k_runs_wo_0, A304442 ptns_w_eq_runsums, A325277 red_topri_seq, A333381 stc_co_runs, A333755 tri_comps_runs, A353844 h_runsum_traj_last_prime, A353848 stc_w_eq_runsums, A353849 stc_numdstnct_runsums, A353850 comps_w_dstnct_runsums, A353852 stc_w_dsntct_runsums, A353855 stc_runsum_traj_last, A353858 comps_runsum_traj_last_sing.

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Gus Wiseman, Jun 04 2022

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