A284398 - OEIS

%I #26 Dec 26 2024 11:09:05

%S 1,4,2,2,0,21,2,30,2,16,1,10,1,2,0,5,0,143,7,260,1,94,1,170,7,20,0,92,

%T 0,5,4,47,0,17,0,11,1,0,0,16,0,0,1,1,0,1,0,1,0,1061,14,2316,1,654,0,

%U 1934,24,128,1,943,1,36,11,753,0,142,0,146,4,3,0,433

%N Table read by rows: T(n,k) is the number of n-digit numbers that have exactly k divisors.

%C Rows begin with row 1: [1, 4, 2, 2] for the nine 1-digit numbers 1..9 (of which one (1) has one divisor, four (the primes: 2, 3, 5, and 7) have two, two (2^2 = 4 and 3^2 = 9) have three, and two (2*3 = 6 and 2^3 = 8) have four).

%C The successive rows have lengths 4, 12, 32, 64, 128, 240, ... (A066150).

%H Jon E. Schoenfield, <a href="/A284398/b284398.txt">Table of n, a(n) for n = 1..1696</a> (first 8 rows of table)

%e Table begins:

%e row 1: 1, 4, 2, 2;

%e row 2: 0, 21, 2, 30, 2, 16, 1, 10, 1, 2, 0, 5;

%e row 3: 0, 143, 7, 260, 1, 94, 1, 170, 7, 20, 0, 92, 0, 5, 4, 47, 0, 17, 0, 11, 1, 0, 0, 16, 0, 0, 1, 1, 0, 1, 0, 1;

%e row 4: 0, 1061, 14, 2316, 1, 654, 0, 1934, 24, 128, 1, 943, 1, 36, 11, 753, 0, 142, 0, 146, 4, 3, 0, 433, 1, 0, 6, 29, 0, 43, 0, 129, 1, 0, 1, 80, 0, 0, 0, 36, 0, 7, 0, 0, 3, 0, 0, 45, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2;

%t Table[Block[{t = KeySort[10^n - 1 + PositionIndex@ DivisorSigma[0, #] &@ Range[10^n, 10^(n + 1) - 1]]}, ReplacePart[ConstantArray[0, Max@ Keys@ t], Map[# -> Length@ Lookup[t, #] &, Keys@ t]]], {n, 0, 3}] (* _Michael De Vlieger_, Nov 01 2017 *)

%Y Columns k=1..6 give A000007, A006879, A379566, A379567, A379568, A379569.

%Y Length of n-th row is A066150(n).

%Y Cf. A000005 (number of divisors).

%K nonn,tabf,base,changed

%O 1,2

%A _Jon E. Schoenfield_, Mar 26 2017