OFFSET
1,1
COMMENTS
For n > 1, a(n) is the number of integers in [prime(n-1)^7 .. prime(n)^7 - 1] with prime signature [7], [3,1] or [1,1,1]. - Robert Israel, Apr 21 2026
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..31
EXAMPLE
a(1) = 15 because [1..127] contains 15 numbers 24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 102, 104, 105, 110, 114 with exactly 8 divisors;
a(2) = 408 because [128..2186] contains 408 numbers 128, 130, 135, ..., 2185 with exactly 8 divisors;
a(3) = 16838 because [2187..78124] contains 16838 numbers 2187, 2193, 2198, ..., 78118 with exactly 8 divisors.
MAPLE
f:= proc(n) local t, a, b, p, q, rmin, rmax;
t:= 1;
b:= ithprime(n)^7-1; a:= ithprime(n-1)^7;
p:= 1:
do
p:= nextprime(p);
if p^3 >= b then break fi;
q:= p;
do
q:= nextprime(q);
if p*q^2 >= b then break fi;
rmin:= max(q, a/(p*q)); rmax:= b/(p*q);
t:= t + NumberTheory:-pi(rmax) - NumberTheory:-pi(rmin);
od;
t:= t + NumberTheory:-pi(b/p^3) - NumberTheory:-pi(a/p^3);
if a < p^4 and p^4 <= b then t:= t - 1 fi;
od;
t
end proc:
f(1):= 15:
map(f, [$1..9]); # Robert Israel, Apr 21 2026
PROG
(Magma) [15] cat [#[k: k in [NthPrime(n-1)^7..NthPrime(n)^7-1] | #Divisors(k) eq 8]: n in [2..4]];
(PARI) a(n) = my(nb=0, ka); if (n==1, ka=1, ka=prime(n-1)^7); forfactored(k=ka, prime(n)^7-1, if (numdiv(k[2])==8, nb++)); nb; \\ Michel Marcus, Apr 21 2026
(Python)
from math import isqrt
from sympy import prime, integer_nthroot, primepi, primerange
def A395379(n):
if n == 1: return 15
def f(x):
p = prime(x)**7
q = integer_nthroot(p, 3)[0]+1
return int(sum(primepi(p//(k*m))-b for a, k in enumerate(primerange(q), 1) for b, m in enumerate(primerange(k+1, isqrt(p//k)+1), a+1))+sum(primepi(p//r**3) for r in primerange(q))-primepi(integer_nthroot(p, 4)[0]))
return 1+f(n)-f(n-1) # Chai Wah Wu, Apr 21 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Apr 20 2026
EXTENSIONS
a(5)-a(9) from Michel Marcus, Apr 21 2026
a(10) from Robert Israel, Apr 21 2026
a(11)-a(22) from Chai Wah Wu, Apr 21 2026
a(23) from Daniel Suteu, Apr 22 2026
STATUS
approved