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

1,1

Comments

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

Links

Table of n, a(n) for n=1..42.

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}.

Example

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

Prog

(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

Crossrefs

Cf. A001358, A036454, A113877, A186621.

Sequence in context: A273321 A091077 A245991 * A295021 A194932 A016994

Adjacent sequences: A186634 A186635 A186636 * A186638 A186639 A186640

Keyword

nonn,easy

Author

Jonathan Vos Post, Feb 24 2011

Status

approved