cp's OEIS Frontend

A065120 Highest power of 2 dividing A057335(n).

%I A065120 #54 Jan 04 2024 05:11:30
%S A065120 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 A065120 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 A065120 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 A065120 Highest power of 2 dividing A057335(n).
%C A065120 a(n) appears on row 1 of the array illustrated in A066099.
%C A065120 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 A065120 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 A065120 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 A065120 From _Daniel Starodubtsev_, Aug 05 2021: (Start)
%F A065120 a(n) = A001511(A059894(n) - 2^A000523(n) + 1) for n > 0 with a(0) = 0.
%F A065120 a(2n+1) = a(n), a(2n) = a(n) + A036987(n-1) for n > 1 with a(0) = 0, a(1) = 1. (End)
%e A065120 A057335(7)= 30 and 30 = 2*3*5 so a(7) = 1; A057335(9)= 24 and 24 = 8*3 so a(9) = 3
%e A065120 From _Omar E. Pol_, Aug 30 2013: (Start)
%e A065120 Written as an irregular triangle with row lengths A011782:
%e A065120   0;
%e A065120   1;
%e A065120   2,1;
%e A065120   3,2,1,1;
%e A065120   4,3,2,2,1,1,1,1;
%e A065120   5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
%e A065120   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 A065120   ...
%e A065120 Column 1 is A001477. Row sums give A000225. Row lengths is A011782.
%e A065120 (End)
%t A065120 nmax = 105;
%t A065120 A062050 = Flatten[Table[Range[2^n], {n, 0, Log[2, nmax] // Ceiling}]];
%t A065120 Module[{b}, b[_] = 0;
%t A065120 a[n_] := If[n == 0, 0, With[{t = A062050[[n]]}, b[t] = b[t] + 1]]];
%t A065120 a /@ Range[0, nmax] (* _Jean-François Alcover_, Jan 12 2022 *)
%o A065120 (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 A065120 (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 A065120 Cf. A066099, A062050, A011782.
%Y A065120 Cf. A003602.
%K A065120 easy,nonn
%O A065120 0,3
%A A065120 _Alford Arnold_, Nov 12 2001
%E A065120 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003