Abstract
Let A be a set of positive integers,p(A,n) be the number of partitions ofnwith parts in A, andp(n)=p(N,n). It is proved that the number ofn⩽Nfor whichp(n) is even is ⪢N, while the number ofn⩽Nfor whichp(n) is odd is ⩾N1/2+o(1). Moreover, by using the theory of modular forms, it is proved (by J.-P. Serre) that, for allaandmthe number ofn, such thatn≡a(modm), andn⩽Nfor whichp(n) is even is ⩾cNfor any constantc, andNlarge enough. Further a set A is constructed with the properties thatp(A,n) is even for alln⩾4 and its counting functionA(x) (the number of elements of A not exceedingx) satisfiesA(x)⪢x/logx. Finally, we study the counting function of sets A such that the number of solutions ofa+a′=n,a,a′∈A,a<a′ is never 1 for largen.
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