A365435 Position of A002110(n) in A005117.

1, 2, 5, 19, 129, 1405, 18262, 310347, 5896728, 135624240, 3933101824, 121926157641, 4511267827532, 184961980943493, 7953365180610401, 373808163488684050, 19811832664899731266, 1168898127229083969893, 71302785760974119699474

Offset: 0

Author

Michael De Vlieger, Nov 19 2023

Keywords

• nonn
• hard
• more

Comments

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

Examples

			Let b(n) = A002110(n) and c(n) = A005117(n).
a(0) = 1 since b(0) = 1, and c(1) = 1.
a(1) = 2 since b(1) = 2 = c(2).
a(2) = 5 since b(2) = 6 = c(5).
a(3) = 19 since b(3) = 30 = c(19), etc.
		

Crossrefs

Cf. A002110, A005117, A071172.

Programs

Mathematica

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

PARI

A071172(n) = my(s); forsquarefree(d=1, sqrtint(n), s += n\d[1]^2 * moebius(d)); s; \\ Charles R Greathouse IV at A071172
a(n) = my(p = prod(i = 1, n, prime(i))); A071172(p); \\ Amiram Eldar, Nov 19 2023

Python

from math import isqrt
from sympy import primorial, mobius
def A365435(n):
    if n == 0: return 1
    p = primorial(n)
    return sum(mobius(k)*(p//k**2) for k in range(1,isqrt(p)+1)) # Chai Wah Wu, Aug 12 2024

Extensions

a(8)-a(16) from Amiram Eldar, Nov 19 2023

a(17) from Chai Wah Wu, Aug 12 2024

a(18) from Chai Wah Wu, Aug 13 2024