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A186637 Semiprime powers with special exponents: k^(j-1) where both j and k are arbitrary semiprime numbers.

%I A186637 #12 Sep 12 2024 14:37:30
%S A186637 64,216,729,1000,1024,2744,3375,7776,9261,10648,15625,17576,35937,
%T A186637 39304,42875,54872,59049,59319,65536,97336,100000,117649,132651,
%U A186637 166375,185193,195112,238328,262144,274625,328509,405224,456533,537824,551368,614125,636056,658503,753571,759375,804357,830584,857375
%N A186637 Semiprime powers with special exponents: k^(j-1) where both j and k are arbitrary semiprime numbers.
%C A186637 Semiprime analog of A036454: prime powers with special exponents: q^(p-1) where both p and q are arbitrary prime numbers.
%F A186637 {a(n)} = {A001358(i) ^ A186621(j)}.
%F A186637 {a(n)} = {a^b where a and b are elements of A001358} = {(p*q)^((r*s)-1) for primes p, q, r, s, not necessarily distinct}.
%e A186637 a(1) = smallest semiprime to power of (smallest semiprime - 1) = 4^(4-1) = 4^3 = 64.
%o A186637 (Python)
%o A186637 from math import isqrt
%o A186637 from sympy import primepi, primerange, integer_nthroot, factorint
%o A186637 def A186637(n):
%o A186637     def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
%o A186637     def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p-1)[0]) for p in range(4,x.bit_length()+1) if sum(factorint(p).values())==2))
%o A186637     def bisection(f,kmin=0,kmax=1):
%o A186637         while f(kmax) > kmax: kmax <<= 1
%o A186637         while kmax-kmin > 1:
%o A186637             kmid = kmax+kmin>>1
%o A186637             if f(kmid) <= kmid:
%o A186637                 kmax = kmid
%o A186637             else:
%o A186637                 kmin = kmid
%o A186637         return kmax
%o A186637     return bisection(f,n,n) # _Chai Wah Wu_, Sep 12 2024
%Y A186637 Cf. A001358, A036454, A113877, A186621.
%K A186637 nonn,easy
%O A186637 1,1
%A A186637 _Jonathan Vos Post_, Feb 24 2011