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A061398 Number of squarefree integers between prime(n) and prime(n+1).

%I A061398 #38 Nov 14 2024 12:11:49
%S A061398 0,0,1,1,0,2,0,2,1,1,3,2,1,1,1,3,0,3,2,0,3,1,3,4,0,1,2,0,2,6,2,2,1,5,
%T A061398 0,2,3,2,1,3,0,6,0,2,0,7,8,1,0,2,3,0,3,3,3,3,0,2,1,1,5,7,2,0,1,9,2,4,
%U A061398 0,0,4,3,2,2,2,2,5,2,4,6,0,5,0,4,1,3,4,1,1,2,6,4,1,4,2,2,7,0,8,4,4,3,2,1,2
%N A061398 Number of squarefree integers between prime(n) and prime(n+1).
%H A061398 Harry J. Smith, <a href="/A061398/b061398.txt">Table of n, a(n) for n = 1..1000</a>
%F A061398 a(n) = A013928(A000040(n+1)) - A013928(A000040(n)) - 1. - _Robert Israel_, Jan 06 2017
%F A061398 a(n) = A373198(n) - 1. - _Gus Wiseman_, Nov 06 2024
%e A061398 Between 113 and 127 the 6 squarefree numbers are 114, 115, 118, 119, 122, 123, so a(30)=6.
%e A061398 From _Gus Wiseman_, Nov 06 2024: (Start)
%e A061398 The a(n) squarefree numbers for n = 1..16:
%e A061398   1   2   3   4   5   6   7   8   9   10  11  12  13  14  15  16
%e A061398   ---------------------------------------------------------------
%e A061398   .   .   6   10  .   14  .   21  26  30  33  38  42  46  51  55
%e A061398                       15      22          34  39              57
%e A061398                                           35                  58
%e A061398 (End)
%p A061398 p:= 2:
%p A061398 for n from 1 to 200 do
%p A061398   q:= nextprime(p);
%p A061398 A[n]:= nops(select(numtheory:-issqrfree, [$p+1..q-1]));
%p A061398 p:= q;
%p A061398 od:
%p A061398 seq(A[i],i=1..200); # _Robert Israel_, Jan 06 2017
%t A061398 a[n_] := Count[Range[Prime[n]+1, Prime[n+1]-1], _?SquareFreeQ];
%t A061398 Array[a, 100] (* _Jean-François Alcover_, Feb 28 2019 *)
%t A061398 Count[Range[#[[1]]+1,#[[2]]-1],_?(SquareFreeQ[#]&)]&/@Partition[ Prime[ Range[120]],2,1] (* _Harvey P. Dale_, Oct 14 2021 *)
%o A061398 (PARI) { n=0; q=2; forprime (p=3, prime(1001), a=0; for (i=q+1, p-1, a+=issquarefree(i)); write("b061398.txt", n++, " ", a); q=p ) } \\ _Harry J. Smith_, Jul 22 2009
%o A061398 (PARI) a(n) = my(pp=prime(n)+1); sum(k=pp, nextprime(pp)-1, issquarefree(k)); \\ _Michel Marcus_, Feb 28 2019
%o A061398 (Python)
%o A061398 from math import isqrt
%o A061398 from sympy import mobius, prime, nextprime
%o A061398 def A061398(n):
%o A061398     p = prime(n)
%o A061398     q = nextprime(p)
%o A061398     r = isqrt(p-1)+1
%o A061398     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))-1 # _Chai Wah Wu_, Jun 01 2024
%Y A061398 Cf. A000040, A005117, A013928.
%Y A061398 Cf. A179211. [_Reinhard Zumkeller_, Jul 05 2010]
%Y A061398 Counting all composite numbers (not just squarefree) gives A046933.
%Y A061398 The version for nonsquarefree numbers is A061399.
%Y A061398 Zeros are A068360.
%Y A061398 The version for prime-powers is A080101.
%Y A061398 Partial sums are A337030.
%Y A061398 The version for non-prime-powers is A368748.
%Y A061398 Excluding prime(n+1) from the range gives A373198.
%Y A061398 Ones are A377430.
%Y A061398 Positives are A377431.
%Y A061398 The version for perfect-powers is A377432.
%Y A061398 The version for non-perfect-powers is A377433 + 2.
%Y A061398 For squarefree numbers (A005117) between primes:
%Y A061398 - length is A061398 (this sequence)
%Y A061398 - min is A112926
%Y A061398 - max is A112925
%Y A061398 - sum is A373197
%Y A061398 For squarefree numbers between powers of two:
%Y A061398 - length is A077643 (except initial terms), partial sums A143658
%Y A061398 - min is A372683, difference A373125, indices A372540, firsts of A372475
%Y A061398 - max is A372889, difference A373126
%Y A061398 - sum is A373123
%Y A061398 For primes between powers of two:
%Y A061398 - length is A036378
%Y A061398 - min is A104080 or A014210, indices A372684 (firsts of A035100)
%Y A061398 - max is A014234, difference A013603
%Y A061398 - sum is A293697 (except initial terms)
%Y A061398 Cf. A001223, A049093, A049094, A076259, A077641, A377434, A377436.
%K A061398 nonn
%O A061398 1,6
%A A061398 _Labos Elemer_, Jun 07 2001