# [seqfan] A new and surprisingly hard elementary number theory question

Neil Sloane njasloane at gmail.com
Thu May 4 04:54:16 CEST 2017

```Sequence is A284597:
a(n) is the least number that begins a run of exactly n consecutive
numbers with a nondecreasing number of divisors.

It begins
46, 5, 43, 1, 1613, 241, 17011, 12853, 234613, 376741, 78312721,
125938261, 4019167441, 16586155153, 35237422882, 1296230533473

Only 16 terms are known. Submitted by Fred Schneider in March, and
corrected by Bill McEachen and Giovanni Resta, Apr 26 2017

The words "begins" and "exactly" in the definition are crucial. The
initial values of tau (number of divisors function, A000005) can be
partitioned into non-decreasing runs as follows:
{1, 2, 2, 3}, {2, 4}, {2, 4}, {3, 4}, {2, 6}, {2, 4, 4, 5}, {2, 6},
{2, 6}, {4, 4}, {2, 8}, {3, 4, 4, 6}, {2, 8}, {2, 6}, {4, 4, 4, 9},
{2, 4, 4, 8}, {2, 8}, {2, 6, 6}, {4}, {2, 10},...
>From this we can see that a(1) = 46 (the first singleton), a(2)=5 (the
first pair), a(3)=43, a(4)=1, ... - Bill McEachen and Giovanni Resta,
Apr 26 2017.

Nice problem!
Neil

```