A300509 - OEIS

%I #40 Aug 01 2021 16:51:14

%S 1,2,1,4,1,6,1,25,3,10,1,212,1,27,8,3625,1,1291,1,7687,18,265,1,

%T 629369,4,885,695,365370,1,685360,1,178723829,131,10782,12,311470930,

%U 1,38692,413,6162245368,1,381481569,1,1067082439,139407,513855,1

%N a(n) is the number of numbers in the interval [2^(n-1), 2^n-1] that have exactly n divisors.

%C Number of n-digit binary numbers with exactly n divisors.

%C If p is an odd prime, then the only p-digit binary number having exactly p divisors is 2^(p-1), so a(p) = 1.

%C Only squares have an odd number of divisors, so for odd values of n, a(n) is the number of numbers in the interval [ceiling(sqrt(2^(n-1))), floor(sqrt(2^n-1))] whose squares have exactly n divisors. The next few odd-indexed terms are a(41) = 1, a(43) = 1, a(45) = 139407, a(47) = 1, and a(49) = 8. - _Jon E. Schoenfield_, May 26 2018

%e a(1) = 1 because the only number in the interval [2^(1-1), 2^1 - 1] = [1, 1] having exactly 1 divisor is 1.

%e a(2) = 2 because each of the two numbers in the interval [2^(2-1), 2^2 - 1] = [2, 3] has exactly 2 divisors.

%e a(8) = 25 because the numbers in the interval [2^(8-1), 2^8 - 1] = [128, 255] having exactly 8 divisors are the 1 number of the form p^7 {i.e., 2^7 = 128}, the 8 numbers of the form p^3 * q {135, 136, 152, 184, 189, 232, 248, 250}, and the 16 numbers of the form p*q*r {130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255}; 1 + 8 + 16 = 25.

%o (PARI) a(n) = sum(k=2^(n-1), 2^n-1, numdiv(k)==n); \\ _Michel Marcus_, May 26 2018

%Y Cf. A000005, A284398.

%Y Main diagonal of A346730.

%K nonn,more,hard

%O 1,2

%A _Jon E. Schoenfield_, May 25 2018

%E a(26)-a(38) from _Giovanni Resta_, May 26 2018

%E a(39) from _Jon E. Schoenfield_, May 26 2018

%E a(40)-a(41) from _Giovanni Resta_, May 27 2018

%E a(42)-a(47) from _Jon E. Schoenfield_, May 27 2018