A395380 - OEIS

%I #28 May 01 2026 16:36:50

%S 4,19,160,483,3089,3328,12563,10379,32184,88507,43668,188210,184134,

%T 113384,276419,559933,772460,312726,1129061,932029,525311,1842227,

%U 1471593,2621705,4372435,2610735,1421029,3089490,1675216,3621466,15889018,5569261,9318716,3378641

%N a(1) = 4; for n > 1, a(n) is the number of integers k in [prime(n-1)^8..prime(n)^8 - 1] with exactly 9 divisors.

%H Chai Wah Wu, <a href="/A395380/b395380.txt">Table of n, a(n) for n = 1..412</a>

%e a(1) = 4 because [1..255] contains 4 numbers 36, 100, 196, 225 with exactly 9 divisors;

%e a(2) = 19 because [256..6560] contains 19 numbers 256, 441, 484, 676, 1089, 1156, 12225, 1444, 1521, 2116, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929 with exactly 9 divisors;

%e a(3) = 160 because [6561..390624] contains 160 numbers 6561, 6524, 7225, ..., 388129 with exactly 9 divisors.

%o (Magma) [4] cat [#[k: k in [NthPrime(n-1)^8..NthPrime(n)^8-1] | #Divisors(k) eq 9]: n in [2..4]];

%o (Python)

%o from math import comb

%o from sympy import prime, primepi, primerange

%o def A395380(n):

%o     if n == 1: return 4

%o     def f(x):

%o         s = prime(x)**2

%o         y = s**2

%o         return int(-(t:=primepi(s))-comb(t,2)+sum(primepi(y//k) for k in primerange(1, s+1)))

%o     return 1+f(n)-f(n-1) # _Chai Wah Wu_, Apr 23 2026

%Y Cf. A000005, A030627, A179645, A390951, A394226.

%K nonn

%O 1,1

%A _Juri-Stepan Gerasimov_, Apr 20 2026

%E a(5)-a(34) from _Chai Wah Wu_, Apr 23 2026