cp's OEIS Frontend

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

%I A145522 #41 Jul 22 2025 13:10:42
%S A145522 1,3,2,6,5,4,10,23,12,7,39,9,97,24,164,484,2759,5044,109,32334,114605,
%T A145522 216960,8,14,252,785135,5503557,28,39222428,75703838,548300521,1496,
%U A145522 2063337476,4008153424,29523940595,3858,112174606866,834662735468,11,12216544412251
%N A145522 a(n) is such that A145521(n) = A053810(a(n)).
%C A145522 This sequence is a permutation of the positive integers. It is its own inverse permutation.
%H A145522 Chai Wah Wu, <a href="/A145522/b145522.txt">Table of n, a(n) for n = 1..55</a>
%H A145522 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.
%F A145522 a(n) = Sum_{primes p, 2^p <= A145521(n)} A000720(floor(A145521(n)^(1/p))).
%F A145522 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
%e A145522 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.
%o A145522 (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
%o A145522 (Python)
%o A145522 from itertools import count
%o A145522 from sympy import integer_nthroot, isprime, primepi
%o A145522 def A145522(n):
%o A145522     total = 0
%o A145522     for p in count(2):
%o A145522         if 2**p > A145521(n): break
%o A145522         if isprime(p): total += primepi(integer_nthroot(A145521(n), p)[0])
%o A145522     return total # _Jason Yuen_, Jan 31 2024
%o A145522 (Python)
%o A145522 from math import prod
%o A145522 from sympy import primepi, integer_nthroot, primerange, factorint
%o A145522 def A145522(n):
%o A145522     def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
%o A145522     m, k = n, f(n)
%o A145522     while m != k:
%o A145522         m, k = k, f(k)
%o A145522     a = prod(e**p for p,e in factorint(m).items())
%o A145522     return sum(primepi(integer_nthroot(a, p)[0]) for p in primerange(a.bit_length())) # _Chai Wah Wu_, Aug 10 2024
%Y A145522 Cf. A053810, A145521, A000720, A060967.
%K A145522 nonn
%O A145522 1,2
%A A145522 _Leroy Quet_, Oct 12 2008
%E A145522 a(11)-a(28) from _Ray Chandler_, Nov 01 2008
%E A145522 a(29)-a(32) from _Jinyuan Wang_, Feb 25 2020
%E A145522 a(33)-a(39) from _Jason Yuen_, Jan 31 2024
%E A145522 a(40) from _Chai Wah Wu_, Aug 10 2024