cp's OEIS Frontend

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.

%I A323743 #9 May 01 2019 05:53:25
%S A323743 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,
%T A323743 21,22,23,24,25,26,27,16,19,20,21,22,23,25,26,27,29,30,31,20,22,24,27,
%U A323743 29,30,31,33,34,35,36,37,38,39
%N 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.
%C A323743 Row n lists the numbers k such that
%C A323743   0 < |{m : Sum_j={m..m+n-1} tau(j) = k}| < infinity
%C A323743 where tau(j) = A000005(j) is the number of divisors of j.
%e A323743 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.
%e A323743 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}.
%e A323743 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.
%e A323743 The first six rows of the table are as follows:
%e A323743   row 1: {1};
%e A323743   row 2: {3, 4, 5};
%e A323743   row 3: {5, 7, 8, 9};
%e A323743   row 4: {8, 9, 11, 12, 13, 14, 15};
%e A323743   row 5: {10, 13, 15, 17, 18, 19};
%e A323743   row 6: {14, 15, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27}.
%Y A323743 Cf. A000005, A005237, A006558, A048892, A072507, A100366, A119479, A141621, A284596, A284597, A292580, A319037, A319045, A319046.
%K A323743 nonn,tabf,more
%O A323743 1,2
%A A323743 _Jon E. Schoenfield_, Apr 02 2019