%I #54 Jan 04 2024 05:11:30

%S 0,1,2,1,3,2,1,1,4,3,2,2,1,1,1,1,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,6,5,

%T 4,4,3,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,7,6,5,5,

%U 4,4,4,4,3,3,3,3,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1

%N Highest power of 2 dividing A057335(n).

%C a(n) appears on row 1 of the array illustrated in A066099.

%C Except for initial zero, ordinal transform of A062050. After initial zero, n-th chunk consists of n, one n-1, two (n-2)'s, ..., 2^(k-1) (n-k)'s, ..., 2^(n-1) 1's. - _Franklin T. Adams-Watters_, Sep 11 2006

%C Zero together with a triangle read by rows in which row j lists the first 2^(j-1) terms of A001511 in nonincreasing order, j >= 1, see example. Also row j lists the first parts, in nonincreasing order, of the compositions of j. - _Omar E. Pol_, Sep 11 2013

%C The n-th row represents the frequency distribution of 1, 2, 3, ..., 2^(n-1) in the first 2^n - 1 terms of A003602. - _Gary W. Adamson_, Jun 10 2021

%F From _Daniel Starodubtsev_, Aug 05 2021: (Start)

%F a(n) = A001511(A059894(n) - 2^A000523(n) + 1) for n > 0 with a(0) = 0.

%F a(2n+1) = a(n), a(2n) = a(n) + A036987(n-1) for n > 1 with a(0) = 0, a(1) = 1. (End)

%e A057335(7)= 30 and 30 = 2*3*5 so a(7) = 1; A057335(9)= 24 and 24 = 8*3 so a(9) = 3

%e From _Omar E. Pol_, Aug 30 2013: (Start)

%e Written as an irregular triangle with row lengths A011782:

%e 0;

%e 1;

%e 2,1;

%e 3,2,1,1;

%e 4,3,2,2,1,1,1,1;

%e 5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;

%e 6,5,4,4,3,3,3,3,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;

%e ...

%e Column 1 is A001477. Row sums give A000225. Row lengths is A011782.

%e (End)

%t nmax = 105;

%t A062050 = Flatten[Table[Range[2^n], {n, 0, Log[2, nmax] // Ceiling}]];

%t Module[{b}, b[_] = 0;

%t a[n_] := If[n == 0, 0, With[{t = A062050[[n]]}, b[t] = b[t] + 1]]];

%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Jan 12 2022 *)

%o (PARI) lista(nn) = {my(v = vector(nn)); v[1] = 1; for (i=2, nn, v[i] = mg(i-1)*v[(i+1)\2];); for (i=1, nn, print1(valuation(v[i], 2), ", "););} \\ _Michel Marcus_, Feb 09 2014

%o (PARI) my(L(n)=if(n,logint(n,2),-1)); a(n) = my(p=L(n)); p - L(n-1<<p); \\ _Kevin Ryde_, Aug 06 2021

%Y Cf. A066099, A062050, A011782.

%Y Cf. A003602.

%K easy,nonn

%O 0,3

%A _Alford Arnold_, Nov 12 2001

%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003