George Andrews kindly informs me that any fixed power of 2 divides almost all Q(n) (usually written q(n), to confuse you with the nome). This is the p=2, k=2 case of the more general result (www.math.wisc.edu/~ono/reprints/018.pdf) that as N ->oo, there is a positive alpha(k,p,j) such that for n<N, p^j fails to divide at most N/(log N)^alpha of the partitions of n into nonmultiples of k, provided that there is an integer a such that p^a|k and p^(2a) > k. While impressive, this can't be sharp, at least for k=2, j=1, p=2, where the density is ~ sqrt(2N/3). On the other hand, for j=2, 325 of the first 801 Q(n) are nonzero mod 4. But even knowing alpha won't tell us much about early appearances of large 2^j. This article (by B. Gordon and K. Ono) cites K. Alladi, www.ams.org/tran/1997-349-12/ S0002-9947-97-01831-X/S0002-9947-97-01831-X.pdf which apparently gives combinatorial arguments for the first few j. He also claims eventual evenness for partitions with even parts distinct. But J. McKay wonders about an odd # of even parts. I get g.f.s 2 3 4 5 6 7 8 9 10 11 12 q + q + 2 q + 3 q + 5 q + 7 q + 10 q + 14 q + 20 q + 27 q + 37 q 13 14 15 16 17 18 19 20 + 49 q + 66 q + 86 q + 113 q + 146 q + 190 q + 242 q + 310 q 21 22 23 24 25 26 27 + 392 q + 497 q + 623 q + 782 q + 973 q + 1212 q + 1498 q 28 29 30 31 32 33 34 + 1851 q + 2274 q + 2793 q + 3411 q + 4163 q + 5059 q + 6142 q 35 36 37 38 39 40 + 7427 q + 8972 q + 10801 q + 12989 q + 15572 q + 18646 q 41 42 43 44 45 46 + 22267 q + 26561 q + 31602 q + 37556 q + 44533 q + 52743 q 47 48 49 50 + 62338 q + 73593 q + 86716 q + 102064 q + . . . 1 1 1 2 -------- - ---------- -------- - (- q; q ) (q; q) (q; - q) (q; q) oo oo oo oo = ----------------------- = ---------------------- 2 2 =(unrestricted - distinct odd)/2 oo oo 2 ==== k ==== k \ q \ q > ------- - > --------- / (q; q) / 2 2 ==== k ==== (q ; q ) k = 0 k = 0 k = ------------------------------- 2 2 8 18 32 50 q - q + q - q + q + . . . = ----------------------------------- (q; q) oo = (electron shell #s = (theta_4(q^2)-1)/2) * unrestricted . Since zilch is known about unrestricted mod 2, this quashes oddevens(n). Confession: I had actually downloaded the Ono article prior to the first even Q(n) message, but suffered mental indigestion due to a defishency. --rwg