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

64, 216, 729, 1000, 1024, 2744, 3375, 7776, 9261, 10648, 15625, 17576, 35937, 39304, 42875, 54872, 59049, 59319, 65536, 97336, 100000, 117649, 132651, 166375, 185193, 195112, 238328, 262144, 274625, 328509, 405224, 456533, 537824, 551368, 614125, 636056, 658503, 753571, 759375, 804357, 830584, 857375

Offset: 1

Author

Jonathan Vos Post, Feb 24 2011

Keywords

• nonn
• easy

Comments

Semiprime analog of A036454: prime powers with special exponents: q^(p-1) where both p and q are arbitrary prime numbers.

Examples

			a(1) = smallest semiprime to power of (smallest semiprime - 1) = 4^(4-1) = 4^3 = 64.
		

Crossrefs

Cf. A001358, A036454, A113877, A186621.

Programs

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A186637(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-1)[0]) for p in range(4,x.bit_length()+1) 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

Formula

{a(n)} = {A001358(i) ^ A186621(j)}.

{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}.