A217908 Semiprime powers of distinct semiprimes.

1296, 4096, 6561, 10000, 38416, 50625, 194481, 234256, 262144, 390625, 456976, 531441, 1000000, 1048576, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 5764801, 6765201, 7529536, 9150625, 10077696, 10556001, 11316496, 11390625, 14776336, 17850625

Offset: 1

Author

Kevin L. Schwartz and Christian N. K. Anderson, Mar 25 2013

Keywords

• nonn

Comments

Subset of A113877.

Examples

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

Links

• Christian N. K. Anderson, Table of n, a(n) for n = 1..9006, for a(n) < 1.5*10^18

Crossrefs

Cf. A113877.

Programs

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A217908(n):
    def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
    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))
    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