OFFSET
1,1
COMMENTS
Triprimes are numbers with exactly three prime factors: A014612.
This is to triprimes as primes are to A053810 (Prime powers of prime numbers) and as semiprimes are to A113877 (Semiprimes to semiprime powers). - Jonathan Vos Post, Mar 26 2013
a(n) increases roughly as n^8, because 9669 of the first 10000 terms are powers of 8. - Kevin L. Schwartz and Christian N. K. Anderson, Jun 05 2013
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000 terms 1..1000 from Kevin L. Schwartz and Christian N. K. Anderson
EXAMPLE
429981696 = 8^12.
a(10) = 9682651996416 = 42^8 = (2*3*7)^(2*2*2).
PROG
(R) library(gmp); istriprime=function(x) ifelse(as.bigz(x)<8, F, length(factorize(x))==3)as.bigz(which(sapply(1:200, istriprime)))->trp; maxy=tail(trp, 1)^trp[1]; len=0; y=as.bigz(rep(0, 100))
for(i in 1:length(trp)) { j=0; while((n=trp[i]^trp[(j=j+1)])<=maxy) y[(len=len+1)]=n }
y[1:len]->y; y[order(as.numeric(y))]
-- Kevin L. Schwartz and Christian N. K. Anderson, Jun 05 2013
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A217784(n):
def g(x): return int(sum(primepi(x//(k*m))-b for a, k in enumerate(primerange(integer_nthroot(x, 3)[0]+1)) for b, m in enumerate(primerange(k, isqrt(x//k)+1), a)))
def f(x): return int(n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length()) if sum(factorint(k).values())==3))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
return bisection(f, n, n) # Chai Wah Wu, Sep 12 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Kevin L. Schwartz and Christian N. K. Anderson, Mar 24 2013
STATUS
approved