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Statement

Setting (p. 259). $\lambda(x)$ is the least positive integer that does not occur among the numbers $d(n)$, $1\le n\le x$.

Theorem VI (p. 259). For $x\ge6$, $\lambda(x)$ is equal to the least prime $q$ satisfying $2^{q-1}>x$.

A footnote on p. 271 records that $\lambda(x)=5$ for $6\le x<16$ and $\lambda(16)=7$.

Source. P. Erdős and L. Mirsky, The distribution of values of the divisor function $d(n)$, Proc. London Math. Soc. (3) 2 (1952), 257--271; Theorem VI on p. 259, its proof in §12, p. 271. The copy read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page images and the proof was read; the small cases the paper checks directly were not re-checked. Nothing here is independently reviewed.

Proof pointer

§12, p. 271. The cases $6\le x\le16$ are checked directly. For $x>16$ let $q'$ be the prime before $q$, so $2^{q'-1}\le x<2^{q-1}$. The value $q$ is missed, since the least integer with $q$ divisors is $2^{q-1}>x$, and every $m\le q'$ occurs. For composite $m=ab$ with $q'<m<q$ and $a\ge b\ge2$, the integer $2^{a-1}3^{b-1}$ has $m$ divisors, and it suffices that $2^{a-1}3^{b-1}\le2^{q'-1}$ (12.1); Bertrand's postulate ($q\le2q'-2$) gives this for $b=2$ and, with an elementary estimate, for $b>2$ and $q'\ge23$, while $b>2$ and $5\le q'\le19$ are checked directly.

Dependencies

Bertrand's postulate, cited from Landau's Handbuch (1909), §22.

Bears on

No Erdős problem in the corpus.