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 A071403 #35 Jul 18 2025 08:34:41 %S A071403 2,3,4,6,8,9,12,13,16,18,20,24,27,29,31,33,37,38,42,45,46,50,52,56,61, %T A071403 62,64,67,68,71,78,81,84,86,92,93,96,100,103,105,109,110,117,118,121, %U A071403 122,130,139,141,142,145,149,150,154,158,162,166,167,170,172,174,180 %N A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime. %C A071403 Also the number of squarefree numbers <= prime(n). - _Gus Wiseman_, Dec 08 2024 %H A071403 Charles R Greathouse IV, <a href="/A071403/b071403.txt">Table of n, a(n) for n = 1..10000</a> %F A071403 A005117(a(n)) = A000040(n) = prime(n). %F A071403 a(n) ~ (6/Pi^2) * n log n. - _Charles R Greathouse IV_, Nov 27 2017 %F A071403 a(n) = A013928(A008864(n)). - _Ridouane Oudra_, Oct 15 2019 %F A071403 From _Gus Wiseman_, Dec 08 2024: (Start) %F A071403 a(n) = A112929(n) + 1. %F A071403 a(n+1) - a(n) = A373198(n) = A061398(n) - 1. %F A071403 (End) %e A071403 a(25)=61 because A005117(61) = prime(25) = 97. %e A071403 From _Gus Wiseman_, Dec 08 2024: (Start) %e A071403 The squarefree numbers up to prime(n) begin: %e A071403 n = 1 2 3 4 5 6 7 8 9 10 %e A071403 ---------------------------------- %e A071403 2 3 5 7 11 13 17 19 23 29 %e A071403 1 2 3 6 10 11 15 17 22 26 %e A071403 1 2 5 7 10 14 15 21 23 %e A071403 1 3 6 7 13 14 19 22 %e A071403 2 5 6 11 13 17 21 %e A071403 1 3 5 10 11 15 19 %e A071403 2 3 7 10 14 17 %e A071403 1 2 6 7 13 15 %e A071403 1 5 6 11 14 %e A071403 3 5 10 13 %e A071403 2 3 7 11 %e A071403 1 2 6 10 %e A071403 1 5 7 %e A071403 3 6 %e A071403 2 5 %e A071403 1 3 %e A071403 2 %e A071403 1 %e A071403 The column-lengths are a(n). %e A071403 (End) %t A071403 Position[Select[Range[300], SquareFreeQ], _?PrimeQ][[All, 1]] (* _Michael De Vlieger_, Aug 17 2023 *) %o A071403 (PARI) lista(nn)=sqfs = select(n->issquarefree(n), vector(nn, i, i)); for (i = 1, #sqfs, if (isprime(sqfs[i]), print1(i, ", "));); \\ _Michel Marcus_, Sep 11 2013 %o A071403 (PARI) a(n,p=prime(n))=sum(k=1, sqrtint(p), p\k^2*moebius(k)) \\ _Charles R Greathouse IV_, Sep 13 2013 %o A071403 (PARI) a(n,p=prime(n))=my(s); forfactored(k=1, sqrtint(p), s+=p\k[1]^2*moebius(k)); s \\ _Charles R Greathouse IV_, Nov 27 2017 %o A071403 (PARI) first(n)=my(v=vector(n),pr,k); forsquarefree(m=1,n*logint(n,2)+3, k++; if(m[2][,2]==[1]~, v[pr++]=k; if(pr==n, return(v)))) \\ _Charles R Greathouse IV_, Jan 08 2018 %o A071403 (Python) %o A071403 from math import isqrt %o A071403 from sympy import prime, mobius %o A071403 def A071403(n): return (p:=prime(n))+sum(mobius(k)*(p//k**2) for k in range(2,isqrt(p)+1)) # _Chai Wah Wu_, Jul 20 2024 %Y A071403 Cf. A000290, A013928. %Y A071403 The strict version is A112929. %Y A071403 A000040 lists the primes, differences A001223, seconds A036263. %Y A071403 A005117 lists the squarefree numbers, differences A076259. %Y A071403 A013929 lists the nonsquarefree numbers, differences A078147. %Y A071403 A070321 gives the greatest squarefree number up to n. %Y A071403 Other families: A014689, A027883, A378615, A065890. %Y A071403 Squarefree numbers between primes: A061398, A068360, A373197, A373198, A377430, A112925, A112926. %Y A071403 Nonsquarefree numbers: A057627, A378086, A061399, A068361, A120327, A377783, A378032, A378033. %Y A071403 Cf. A046933, A049093, A053797, A072284, A077641, A224363, A337030, A345531, A008864. %K A071403 nonn %O A071403 1,1 %A A071403 _Labos Elemer_, May 24 2002