### 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 ≥N^{1/2+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.

Original language | English |
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Pages (from-to) | 292-317 |

Number of pages | 26 |

Journal | Journal of Number Theory |

Volume | 73 |

Issue number | 2 |

DOIs | |

Publication status | Published - Dec 1998 |

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Number Theory*,

*73*(2), 292-317. https://doi.org/10.1006/jnth.1998.2288