cp's OEIS Frontend

A380337 Number of perfect powers (in A001597) that do not exceed primorial A002110(n).

%I A380337 #22 Jan 23 2025 15:34:29
%S A380337 1,1,2,7,19,63,208,802,3344,15576,82368,453834,2743903,17510668,
%T A380337 114616907,785002449,5711892439,43861741799,342522899289,
%U A380337 2803468693325,23621594605383,201819398349092,1793794228847381,16342173067958793,154171432351500060,1518411003599957803
%N A380337 Number of perfect powers (in A001597) that do not exceed primorial A002110(n).
%C A380337 In other words, A001597(a(n)) is the largest perfect power less than or equal to A002110(n).
%H A380337 Michael De Vlieger, <a href="/A380337/b380337.txt">Table of n, a(n) for n = 0..632</a>
%e A380337 Let P = A002110 and let s = A001597.
%e A380337 a(0) = 1 since P(0) = 1, and the set s(1) = {1} contains k that do not exceed 1.
%e A380337 a(1) = 1 since P(1) = 2, and the set s(1) = {1} contains k <= 2.
%e A380337 a(2) = 2 since P(2) = 6, and the set s(1..2) = {1, 4} contains k <= 6.
%e A380337 a(3) = 7 since P(3) = 30, and the set s(1..7) = {1, 4, 8, 9, 16, 25, 27} contains k <= 30.
%e A380337 a(4) = 19 since P(4) = 210, and the set s(1..19) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196} contains k <= 210, etc.
%t A380337 Map[1 - Sum[MoebiusMu[k]*Floor[#^(1/k) - 1], {k, 2, Floor[Log2[#]]}] &, FoldList[Times, 1, Prime[Range[30]]] ]
%o A380337 (Python)
%o A380337 from sympy import primorial, mobius, integer_nthroot
%o A380337 def A380337(n):
%o A380337     if n == 0: return 1
%o A380337     p = primorial(n)
%o A380337     return int(1-sum(mobius(k)*(integer_nthroot(p,k)[0]-1) for k in range(2,p.bit_length()))) # _Chai Wah Wu_, Jan 23 2025
%Y A380337 Cf. A001597, A002110, A070228, A070428, A380254.
%K A380337 nonn
%O A380337 0,3
%A A380337 _Michael De Vlieger_, Jan 21 2025