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

1,1

Comments

Subset of A113877.

Links

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

Example

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

Prog

(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

Crossrefs

Cf. A113877.

Sequence in context: A281399 A043372 A372841 * A223508 A250810 A378900

Adjacent sequences: A217905 A217906 A217907 * A217909 A217910 A217911

Keyword

nonn

Author

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

Status

approved