A379566
Number of n-digit numbers that have exactly 3 divisors.
Original entry on oeis.org
2, 2, 7, 14, 40, 103, 278, 783, 2172, 6191, 17701, 51205, 149149, 436932, 1287378, 3809498, 11321211, 33764868, 101029398, 303175579, 912147300, 2750855002, 8313825647, 25176031558, 76375623757, 232082001064, 706304629714, 2152571584249, 6568923555719
Offset: 1
A379568
Number of n-digit numbers that have exactly 5 divisors.
Original entry on oeis.org
0, 2, 1, 1, 3, 4, 5, 9, 15, 25, 37, 66, 107, 171, 293, 490, 810, 1362, 2302, 3889, 6552, 11149, 18950, 32255, 55053, 94096, 161036, 275896, 473709, 813669, 1399593, 2409905, 4154437, 7166774, 12375776, 21389092, 36994679, 64034719, 110918422, 192257157, 333449674, 578697626
Offset: 1
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Table[PrimePi[10^(n/4)]-PrimePi[10^((n-1)/4)],{n,50}] (* Vincenzo Librandi, Dec 30 2024 *)
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from sympy import primepi
def a379568(n): return primepi(10**(n/4)) - primepi(10**((n-1)/4)) # David Radcliffe, Dec 29 2024
A379569
Number of n-digit numbers that have exactly 6 divisors.
Original entry on oeis.org
0, 16, 94, 654, 4863, 38243, 313705, 2658846, 23073712, 203859889, 1826368510, 16544195786, 151222451513, 1392635179004, 12906366376283, 120260052661235
Offset: 1
For n = 2 the a(2) = 16 numbers are 12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99.
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from math import isqrt
from sympy import primerange, primepi, integer_nthroot
def _sum(N): return sum(primepi(N//(p * p)) for p in primerange(isqrt(N//2)+1)) - primepi(integer_nthroot(N, 3)[0]) + primepi(integer_nthroot(N, 5)[0])
def a379569(n): return sum(10**n) - _sum(10**(n-1)) # _David Radcliffe, Dec 29 2024
A300509
a(n) is the number of numbers in the interval [2^(n-1), 2^n-1] that have exactly n divisors.
Original entry on oeis.org
1, 2, 1, 4, 1, 6, 1, 25, 3, 10, 1, 212, 1, 27, 8, 3625, 1, 1291, 1, 7687, 18, 265, 1, 629369, 4, 885, 695, 365370, 1, 685360, 1, 178723829, 131, 10782, 12, 311470930, 1, 38692, 413, 6162245368, 1, 381481569, 1, 1067082439, 139407, 513855, 1
Offset: 1
a(1) = 1 because the only number in the interval [2^(1-1), 2^1 - 1] = [1, 1] having exactly 1 divisor is 1.
a(2) = 2 because each of the two numbers in the interval [2^(2-1), 2^2 - 1] = [2, 3] has exactly 2 divisors.
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.
A379567
Number of n-digit numbers that have exactly 4 divisors.
Original entry on oeis.org
2, 30, 260, 2316, 20719, 186565, 1694033, 15522194, 143359184, 1332981873, 12466196499, 117165976234, 1105961883514, 10478813824875, 99613913708218, 949727471728542
Offset: 1
A379570
Number of n-digit numbers that have exactly 8 divisors.
Original entry on oeis.org
0, 10, 170, 1934, 20067, 202246, 2003991, 19674052, 192215670, 1873532828, 18242642732, 177582019015, 1728951136938, 16840198807124, 164117159854744, 1600427660469575, 15617400806292160
Offset: 1
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