A145522 a(n) is such that A145521(n) = A053810(a(n)).

1, 3, 2, 6, 5, 4, 10, 23, 12, 7, 39, 9, 97, 24, 164, 484, 2759, 5044, 109, 32334, 114605, 216960, 8, 14, 252, 785135, 5503557, 28, 39222428, 75703838, 548300521, 1496, 2063337476, 4008153424, 29523940595, 3858, 112174606866, 834662735468, 11, 12216544412251

Offset: 1

Author

Leroy Quet, Oct 12 2008

Keywords

• nonn

Comments

This sequence is a permutation of the positive integers. It is its own inverse permutation.

Examples

			The primes raised to prime exponents form the sequence, when the terms are arranged in numerical order, 4,8,9,25,27,32,49,121,125,128,...(sequence A053810). The 10th term is 128, which is 2^7. So the 10th term of sequence A145521 is 7^2 = 49. 49 is the 7th term of A053810. So a(10) = 7 and a(7) = 10.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..55
• Index entries for sequences that are permutations of the natural numbers.

Crossrefs

Cf. A053810, A145521, A000720, A060967.

Programs

PARI

lista(nn) = {my(c, m); for(k=1, nn, if(isprime(isprimepower(k, &p)), c=0; m=bigomega(k)^p; forprime(q=2, sqrtint(m), c+=primepi(logint(m, q))); print1(c, ", "))); } \\ Jinyuan Wang, Feb 25 2020

Python

from itertools import count
from sympy import integer_nthroot, isprime, primepi
def A145522(n):
    total = 0
    for p in count(2):
        if 2**p > A145521(n): break
        if isprime(p): total += primepi(integer_nthroot(A145521(n), p)[0])
    return total # Jason Yuen, Jan 31 2024

Python

from math import prod
from sympy import primepi, integer_nthroot, primerange, factorint
def A145522(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    a = prod(e**p for p,e in factorint(m).items())
    return sum(primepi(integer_nthroot(a, p)[0]) for p in primerange(a.bit_length())) # Chai Wah Wu, Aug 10 2024

Formula

a(n) = Sum_{primes p, 2^p <= A145521(n)} A000720(floor(A145521(n)^(1/p))).

Also, if A145521(n) = 2^k then a(n) = A060967(k) + Sum_{primes p, 3 <= p <= k} A000720(floor(2^(k/p))). - Jason Yuen, Jan 31 2024

Extensions

a(11)-a(28) from Ray Chandler, Nov 01 2008

a(29)-a(32) from Jinyuan Wang, Feb 25 2020

a(33)-a(39) from Jason Yuen, Jan 31 2024

a(40) from Chai Wah Wu, Aug 10 2024