cp's OEIS Frontend

A373198 Number of squarefree numbers from prime(n) to prime(n+1) - 1.

%I A373198 #10 Jun 01 2024 18:04:28
%S A373198 1,1,2,2,1,3,1,3,2,2,4,3,2,2,2,4,1,4,3,1,4,2,4,5,1,2,3,1,3,7,3,3,2,6,
%T A373198 1,3,4,3,2,4,1,7,1,3,1,8,9,2,1,3,4,1,4,4,4,4,1,3,2,2,6,8,3,1,2,10,3,5,
%U A373198 1,1,5,4,3,3,3,3,6,3,5,7,1,6,1,5,2,4,5
%N A373198 Number of squarefree numbers from prime(n) to prime(n+1) - 1.
%F A373198 a(n) = A061398(n) + 1.
%e A373198 This is the sequence of row-lengths of A005117 treated as a triangle with row-sums A373197:
%e A373198    2
%e A373198    3
%e A373198    5   6
%e A373198    7  10
%e A373198   11
%e A373198   13  14  15
%e A373198   17
%e A373198   19  21  22
%e A373198   23  26
%e A373198   29  30
%e A373198   31  33  34  35
%e A373198   37  38  39
%e A373198   41  42
%e A373198   43  46
%e A373198   47  51
%e A373198   53  55  57  58
%t A373198 Table[Length[Select[Range[Prime[n],Prime[n+1]-1],SquareFreeQ]],{n,100}]
%o A373198 (Python)
%o A373198 from math import isqrt
%o A373198 from sympy import prime, nextprime, mobius
%o A373198 def A373198(n):
%o A373198     p = prime(n)
%o A373198     q = nextprime(p)
%o A373198     r = isqrt(p-1)+1
%o A373198     return sum(mobius(k)*((q-1)//k**2) for k in range(r,isqrt(q-1)+1))+sum(mobius(k)*((q-1)//k**2-(p-1)//k**2) for k in range(1,r)) # _Chai Wah Wu_, Jun 01 2024
%Y A373198 Counting all numbers (not just squarefree) gives A001223, sum A371201.
%Y A373198 For composite instead of squarefree we have A046933.
%Y A373198 For squarefree numbers (A005117) between primes:
%Y A373198 - sum is A373197
%Y A373198 - length is A373198 (this sequence) = A061398 - 1
%Y A373198 - min is A000040
%Y A373198 - max is A112925, opposite A112926
%Y A373198 For squarefree numbers between powers of two:
%Y A373198 - sum is A373123
%Y A373198 - length is A077643, partial sums A143658
%Y A373198 - min is A372683, delta A373125, indices A372540, firsts of A372475
%Y A373198 - max is A372889, delta A373126
%Y A373198 For primes between powers of two:
%Y A373198 - sum is A293697 (except initial terms)
%Y A373198 - length is A036378
%Y A373198 - min is A104080 or A014210, indices A372684 (firsts of A035100)
%Y A373198 - max is A014234, delta A013603
%Y A373198 Cf. A372473 (firsts of A372472), A372541 (firsts of A372433).
%Y A373198 Cf. A049093, A049094, A076259, A077641.
%K A373198 nonn
%O A373198 1,3
%A A373198 _Gus Wiseman_, May 29 2024