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A007149 2-part of number of graphs on n nodes.

%I A007149 M0017 #32 Jul 04 2024 19:59:59
%S A007149 0,0,1,2,0,1,2,2,1,2,4,3,4,4,5,5,4,5,8,6,8,7,8,8,9,9,10,10,15,11,12,
%T A007149 12,11,12,16,13,16,14,15,15,17,16,17,17,19,18,19,19,20,20,21,21,23,22,
%U A007149 23,23,25,24,25,25,27,26,27,27,26,27,31,28,32,29,30,30,35,31,32,32,34,33,34,34,36,35,36,36,38,37,38,38
%N A007149 2-part of number of graphs on n nodes.
%D A007149 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007149 Chai Wah Wu, <a href="/A007149/b007149.txt">Table of n, a(n) for n = 0..97</a>
%H A007149 Steven C. Cater and Robert W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/techtournament.pdf">Exponents of 2 in the numbers of unlabeled graphs and tournaments</a>, Congressus Numerantium, 82 (1991), pp. 139-155.
%H A007149 Steven C. Cater and Robert W. Robinson, <a href="/A007149/a007149.pdf">Exponents of 2 in the numbers of unlabeled graphs and tournaments</a>, Preprint. (Annotated scanned copy)
%F A007149 a(n) = A007814(A000088(n)). - _Michel Marcus_, Jan 06 2020
%t A007149 A000088 = Cases[Import["https://oeis.org/A000088/b000088.txt", "Table"], {_, _}][[All, 2]];
%t A007149 IntegerExponent[#, 2]& /@ A000088 (* _Jean-François Alcover_, Jan 06 2020 *)
%o A007149 (Python)
%o A007149 from itertools import combinations
%o A007149 from math import prod, factorial, gcd
%o A007149 from fractions import Fraction
%o A007149 from sympy.utilities.iterables import partitions
%o A007149 def A007149(n): return (~(m:=int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))) & m-1).bit_length() # _Chai Wah Wu_, Jul 02 2024
%Y A007149 Power of 2 dividing A000088. Cf. A007814.
%K A007149 nonn
%O A007149 0,4
%A A007149 _N. J. A. Sloane_
%E A007149 More terms from _Alois P. Heinz_, Aug 15 2019