A217784 Triprimes to triprime powers.

16777216, 429981696, 11019960576, 25600000000, 68719476736, 282429536481, 377801998336, 656100000000, 8916100448256, 9682651996416, 14048223625216, 16815125390625, 39062500000000, 53459728531456, 248155780267521, 360040606269696, 457163239653376, 576480100000000

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, Mar 24 2013

Keywords

• nonn

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

Examples

			429981696 = 8^12.
a(10) = 9682651996416 = 42^8 = (2*3*7)^(2*2*2).
		

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

Crossrefs

Cf. A014612, A053810, A113877, A129539.

Programs

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

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