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Semiprime powers of distinct semiprimes.
1

%I #18 Sep 12 2024 15:58:35

%S 1296,4096,6561,10000,38416,50625,194481,234256,262144,390625,456976,

%T 531441,1000000,1048576,1185921,1336336,1500625,2085136,2313441,

%U 4477456,5764801,6765201,7529536,9150625,10077696,10556001,11316496,11390625,14776336,17850625

%N Semiprime powers of distinct semiprimes.

%C Subset of A113877.

%H Christian N. K. Anderson, <a href="/A217908/b217908.txt">Table of n, a(n) for n = 1..9006</a>, for a(n) < 1.5*10^18

%e 6561=9^4, and 9 and 4 are both semiprime. 46656 = 6^6 is excluded because the semiprimes are not distinct.

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot, factorint

%o def A217908(n):

%o def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))

%o def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p)[0])-(p**p<=x) for p in range(4,x.bit_length()) if sum(factorint(p).values())==2))

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o return bisection(f,n,n) # _Chai Wah Wu_, Sep 12 2024

%Y Cf. A113877.

%K nonn

%O 1,1

%A _Kevin L. Schwartz_ and _Christian N. K. Anderson_, Mar 25 2013