This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378086 #10 Dec 05 2024 12:06:50
%S A378086 0,0,1,1,3,4,5,6,7,11,11,13,14,14,16,20,22,23,25,26,27,29,31,33,36,39,
%T A378086 39,40,41,42,49,50,53,53,57,58,61,63,64,68,70,71,74,75,76,77,81,84,86,
%U A378086 87,88,90,91,97,99,101,103,104,107,109,109,113,119,120,121
%N A378086 Number of nonsquarefree numbers < prime(n).
%F A378086 a(n) = A057627(prime(n)).
%e A378086 The nonsquarefree numbers counted under each term begin:
%e A378086 n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9: n=10: n=11: n=12:
%e A378086 --------------------------------------------------------------
%e A378086 . . 4 4 9 12 16 18 20 28 28 36
%e A378086 8 9 12 16 18 27 27 32
%e A378086 4 8 9 12 16 25 25 28
%e A378086 4 8 9 12 24 24 27
%e A378086 4 8 9 20 20 25
%e A378086 4 8 18 18 24
%e A378086 4 16 16 20
%e A378086 12 12 18
%e A378086 9 9 16
%e A378086 8 8 12
%e A378086 4 4 9
%e A378086 8
%e A378086 4
%t A378086 Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}]
%o A378086 (Python)
%o A378086 from math import isqrt
%o A378086 from sympy import prime, mobius
%o A378086 def A378086(n): return (p:=prime(n))-sum(mobius(k)*(p//k**2) for k in range(1,isqrt(p)+1)) # _Chai Wah Wu_, Dec 05 2024
%Y A378086 For nonprime numbers we have A014689.
%Y A378086 Restriction of A057627 to the primes.
%Y A378086 First-differences are A061399 (zeros A068361), squarefree A061398 (zeros A068360).
%Y A378086 For composite instead of squarefree we have A065890.
%Y A378086 For squarefree we have A071403, differences A373198.
%Y A378086 Greatest is A378032 (differences A378034), restriction of A378033 (differences A378036).
%Y A378086 A000040 lists the primes, differences A001223, seconds A036263.
%Y A378086 A005117 lists the squarefree numbers.
%Y A378086 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A378086 A070321 gives the greatest squarefree number up to n.
%Y A378086 A112925 gives the greatest squarefree number between primes, differences A378038.
%Y A378086 A112926 gives the least squarefree number between primes, differences A378037.
%Y A378086 A120327 gives the least nonsquarefree number >= n, first-differences A378039.
%Y A378086 A377783 gives the least nonsquarefree > prime(n), differences A377784.
%Y A378086 Cf. A013928, A046933, A053797, A053806, A072284, A076259, A224363, A337030, A377049, A378040, A378082, A378084.
%K A378086 nonn
%O A378086 1,5
%A A378086 _Gus Wiseman_, Dec 04 2024