A145521 Take the primes raised to prime exponents, arranged in numerical order (A053810). If A053810(n) = r(n)^q(n), where r(n) and q(n) are primes, then a(n) = q(n)^r(n).

4, 9, 8, 32, 27, 25, 128, 2048, 243, 49, 8192, 125, 131072, 2187, 524288, 8388608, 536870912, 2147483648, 177147, 137438953472, 2199023255552, 8796093022208, 121, 343, 1594323, 140737488355328, 9007199254740992, 3125, 576460752303423488, 2305843009213693952, 147573952589676412928

Offset: 1

Author

Leroy Quet, Oct 12 2008

Keywords

• nonn

Comments

a(n) = A053812(n)^A053811(n).

Crossrefs

Cf. A053810, A053811, A053812, A145522.

Programs

PARI

lista(nn) = for(k=1, nn, if(isprime(isprimepower(k, &p)), print1(bigomega(k)^p, ", "))); \\ Jinyuan Wang, Feb 25 2020

Python

from math import prod
from sympy import primepi, integer_nthroot, primerange, factorint
def A145521(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return prod(e**p for p,e in factorint(kmax).items()) # Chai Wah Wu, Aug 13 2024

Extensions

Extended by Ray Chandler, Nov 01 2008

More terms from Jinyuan Wang, Feb 25 2020