A365435 - OEIS

%I #29 Aug 13 2024 21:03:54

%S 1,2,5,19,129,1405,18262,310347,5896728,135624240,3933101824,

%T 121926157641,4511267827532,184961980943493,7953365180610401,

%U 373808163488684050,19811832664899731266,1168898127229083969893,71302785760974119699474

%N Position of A002110(n) in A005117.

%C Primorial A002110(n) is the a(n)-th squarefree number.

%e Let b(n) = A002110(n) and c(n) = A005117(n).

%e a(0) = 1 since b(0) = 1, and c(1) = 1.

%e a(1) = 2 since b(1) = 2 = c(2).

%e a(2) = 5 since b(2) = 6 = c(5).

%e a(3) = 19 since b(3) = 30 = c(19), etc.

%t s = Select[Range[2^20], SquareFreeQ]; Map[FirstPosition[s, #][[1]] &, FoldList[Times, 1, Prime@ Range[7]] ]

%o (PARI) A071172(n) = my(s); forsquarefree(d=1, sqrtint(n), s += n\d[1]^2 * moebius(d)); s; \\ _Charles R Greathouse IV_ at A071172

%o a(n) = my(p = prod(i = 1, n, prime(i))); A071172(p); \\ _Amiram Eldar_, Nov 19 2023

%o (Python)

%o from math import isqrt

%o from sympy import primorial, mobius

%o def A365435(n):

%o     if n == 0: return 1

%o     p = primorial(n)

%o     return sum(mobius(k)*(p//k**2) for k in range(1,isqrt(p)+1)) # _Chai Wah Wu_, Aug 12 2024

%Y Cf. A002110, A005117, A071172.

%K nonn,hard,more

%O 0,2

%A _Michael De Vlieger_, Nov 19 2023

%E a(8)-a(16) from _Amiram Eldar_, Nov 19 2023

%E a(17) from _Chai Wah Wu_, Aug 12 2024

%E a(18) from _Chai Wah Wu_, Aug 13 2024