A323743 Table read by rows: row n lists the numbers k for which there exist only finitely many runs of n consecutive integers whose number-of-divisors function sums to k.

1, 3, 4, 5, 5, 7, 8, 9, 8, 9, 11, 12, 13, 14, 15, 10, 13, 15, 17, 18, 19, 14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 16, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 20, 22, 24, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39

Offset

1,2

Comments

Row n lists the numbers k such that

0 < |{m : Sum_j={m..m+n-1} tau(j) = k}| < infinity

where tau(j) = A000005(j) is the number of divisors of j.

Links

Table of n, a(n) for n=1..59.

Example

There is only one number with exactly 1 divisor (namely, k=1), but there are infinitely many numbers with j divisors for every j >= 2, so row 1 consists only of the single term 1.
The sequence of values tau(k) for k >= 1 is A000005, which begins 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ..., from which the sums of two consecutive terms are 1+2=3, 2+2=4, 2+3=5, 3+2=5, 2+4=6, 4+2=6, 2+4=6, 4+3=7, 3+4=7, ...; no number j < 3 appears as such a sum, every j >= 6 appears infinitely many times as such a sum, and each j in {3,4,5} appears as such a sum only finitely many times, so row 2 is {3, 4, 5}.
Row 3 does not contain 6 as a term because there exists no run of 3 consecutive numbers whose sum of tau values is exactly 6.
The first six rows of the table are as follows:
  row 1: {1};
  row 2: {3, 4, 5};
  row 3: {5, 7, 8, 9};
  row 4: {8, 9, 11, 12, 13, 14, 15};
  row 5: {10, 13, 15, 17, 18, 19};
  row 6: {14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27}.

Crossrefs

Cf. A000005, A005237, A006558, A048892, A072507, A100366, A119479, A141621, A284596, A284597, A292580, A319037, A319045, A319046.

Sequence in context: A082514 A227215 A229445 * A261017 A024357 A300778

Adjacent sequences: A323740 A323741 A323742 * A323744 A323745 A323746

Keyword

nonn,tabf,more

Author

Jon E. Schoenfield, Apr 02 2019

Status

approved