Problem 1106

Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of

\prod_{1\leq k\leq n}p(k).

Does $F(n)\to \infty$ with $n$ ? Is $F(n)>n$ for all sufficiently large $n$ ?

Source: erdosproblems.com/1106

Number theory

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Open

No claim settles this problem.

Open, the site's label (OPEN, page last edited 16 November 2025). The first question is answered yes (Schinzel, with the proof in [ErIv90]; [ScWi87]; [On00]); the second, whether $F(n)>n$ for all large $n$ , is open. The page lists the two parts as tends_to_infinity and exceeds_n.

ErIv90: Erdős, Paul and Ivić, Aleksandar, The distribution of values of a certain class of arithmetic functions at consecutive integers. Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai 51 (1990), 45-91.
Ob1: P. Erdős, Oberwolfach Mathematical Problems, Volume 1, Mathematisches Forschungsinstitut Oberwolfach; the site's source for the problem.
On00: Ono, Ken, Distribution of the partition function modulo $m$ . Ann. of Math. (2) (2000), 293-307.
ScWi87: Schinzel, A. and Wirsing, E., Multiplicative properties of the partition function. Proc. Indian Acad. Sci. Math. Sci. 97 (1987), nos. 1-3, 297-303; doi:10.1007/BF02837831.
Ti73: Tijdeman, R., On integers with many small prime factors. Compositio Math. (1973), 319-330.

Problem 1106

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