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A217784 Triprimes to triprime powers.

%I A217784 #36 Sep 12 2024 15:03:27
%S A217784 16777216,429981696,11019960576,25600000000,68719476736,282429536481,
%T A217784 377801998336,656100000000,8916100448256,9682651996416,14048223625216,
%U A217784 16815125390625,39062500000000,53459728531456,248155780267521,360040606269696,457163239653376,576480100000000
%N A217784 Triprimes to triprime powers.
%C A217784 Triprimes are numbers with exactly three prime factors: A014612.
%C A217784 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
%C A217784 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
%H A217784 Chai Wah Wu, <a href="/A217784/b217784.txt">Table of n, a(n) for n = 1..10000</a> terms 1..1000 from Kevin L. Schwartz and Christian N. K. Anderson
%e A217784 429981696 = 8^12.
%e A217784 a(10) = 9682651996416 = 42^8 = (2*3*7)^(2*2*2).
%o A217784 (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))
%o A217784 for(i in 1:length(trp)) { j=0; while((n=trp[i]^trp[(j=j+1)])<=maxy) y[(len=len+1)]=n }
%o A217784 y[1:len]->y; y[order(as.numeric(y))]
%o A217784 -- _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Jun 05 2013
%o A217784 (Python)
%o A217784 from math import isqrt
%o A217784 from sympy import primepi, primerange, integer_nthroot, factorint
%o A217784 def A217784(n):
%o A217784     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)))
%o A217784     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))
%o A217784     def bisection(f,kmin=0,kmax=1):
%o A217784         while f(kmax) > kmax: kmax <<= 1
%o A217784         while kmax-kmin > 1:
%o A217784             kmid = kmax+kmin>>1
%o A217784             if f(kmid) <= kmid:
%o A217784                 kmax = kmid
%o A217784             else:
%o A217784                 kmin = kmid
%o A217784         return kmax
%o A217784     return bisection(f,n,n) # _Chai Wah Wu_, Sep 12 2024
%Y A217784 Cf. A014612, A053810, A113877, A129539.
%K A217784 nonn
%O A217784 1,1
%A A217784 _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Mar 24 2013