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A112926 Smallest squarefree integer > the n-th prime.

%I A112926 #26 Dec 09 2024 22:04:15
%S A112926 3,5,6,10,13,14,19,21,26,30,33,38,42,46,51,55,61,62,69,73,74,82,85,91,
%T A112926 101,102,105,109,110,114,129,133,138,141,151,154,158,165,170,174,181,
%U A112926 182,193,194,199,201,213,226,229,230,235,241,246,253,258,265,271,273
%N A112926 Smallest squarefree integer > the n-th prime.
%H A112926 Michael De Vlieger, <a href="/A112926/b112926.txt">Table of n, a(n) for n = 1..10000</a>
%F A112926 a(n) = prime(n) + A240474(n). - _Gus Wiseman_, Dec 07 2024
%e A112926 10 is the smallest squarefree number greater than the 4th prime, 7. So a(4) = 10.
%e A112926 From _Gus Wiseman_, Dec 07 2024: (Start)
%e A112926 The first number line below shows the squarefree numbers. The second shows the primes:
%e A112926 --1--2--3-----5--6--7-------10-11----13-14-15----17----19----21-22-23-------26--
%e A112926 =====2==3=====5=====7==========11====13==========17====19==========23===========
%e A112926 (End)
%p A112926 with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from p+1 to p+20 do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: B[1] end: seq(a(m),m=1..75); # _Emeric Deutsch_, Oct 10 2005
%t A112926 Do[k = Prime[n] + 1; While[ !SquareFreeQ[k], k++ ]; Print[k], {n, 1, 100}] (* _Ryan Propper_, Oct 10 2005 *)
%t A112926 With[{k = 120}, Table[SelectFirst[Range[Prime@ n + 1, Prime@ n + k], SquareFreeQ], {n, 58}]] (* _Michael De Vlieger_, Aug 16 2017 *)
%o A112926 (PARI) a(n,p=prime(n))=while(!issquarefree(p++),); p \\ _Charles R Greathouse IV_, Aug 16 2017
%Y A112926 Cf. A061400, A112928, A112929, A112930.
%Y A112926 Restriction of A067535, differences A378087.
%Y A112926 The unrestricted opposite is A070321, differences A378085.
%Y A112926 The opposite is A112925, differences A378038.
%Y A112926 Subtracting prime(n) from each term gives A240474, opposite A240473.
%Y A112926 For nonsquarefree we have A377783, restriction of A120327.
%Y A112926 The nonsquarefree differences are A377784, restriction of A378039.
%Y A112926 First differences are A378037.
%Y A112926 For perfect power we have A378249, A378617, A378250, A378251.
%Y A112926 A000040 lists the primes, differences A001223, seconds A036263.
%Y A112926 A005117 lists the squarefree numbers.
%Y A112926 A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
%Y A112926 A061398 counts squarefree numbers between primes, zeros A068360.
%Y A112926 A061399 counts nonsquarefree numbers between primes, zeros A068361.
%Y A112926 Cf. A007674, A013928, A053797, A053806, A072284, A073247, A175216, A280892, A345531, A378032, A378618.
%K A112926 nonn
%O A112926 1,1
%A A112926 _Leroy Quet_, Oct 06 2005
%E A112926 More terms from _Ryan Propper_ and _Emeric Deutsch_, Oct 10 2005