A047972 - OEIS

%I #36 Apr 27 2022 19:06:00

%S 1,1,1,2,2,3,1,3,2,4,5,1,5,6,2,4,5,3,3,7,8,2,2,8,3,1,3,7,9,8,6,10,7,5,

%T 5,7,12,6,2,4,10,12,5,3,1,3,14,2,2,4,8,14,15,5,1,7,13,15,12,8,6,4,17,

%U 13,11,7,7,13,14,12,8,2,6,12,18,17,11,3,1,9,19,20,10,8,2,2,8,16,20,21

%N Distance of n-th prime to nearest square.

%C Conjecture: a(n) < sqrt(prime(n)) and lim_{n->infinity} a(n)/sqrt(prime(n)) = 1, where prime(n) is the n-th prime. - _Ya-Ping Lu_, Nov 29 2021

%H T. D. Noe, <a href="/A047972/b047972.txt">Table of n, a(n) for n = 1..1000</a>

%F For each prime, find the closest square (preceding or succeeding); subtract, take absolute value.

%e For 13, 9 is the preceding square, 16 is the succeeding. 13-9 = 4, 16-13 is 3, so the distance is 3.

%t a[n_] := (p = Prime[n]; ns = p+1; While[ !IntegerQ[ Sqrt[ns++]]]; ps = p-1; While[ !IntegerQ[ Sqrt[ps--]]]; Min[ ns-p-1, p-ps-1]); Table[a[n], {n, 1, 90}] (* _Jean-François Alcover_, Nov 18 2011 *)

%t Table[Apply[Min, Abs[p - Through[{Floor, Ceiling}[Sqrt[p]]]^2]], {p, Prime[Range[90]]}] (* _Jan Mangaldan_, May 07 2014 *)

%t Min[Abs[#-Through[{Floor,Ceiling}[Sqrt[#]]]^2]]&/@Prime[Range[100]] (* More concise version of program immediately above *) (* _Harvey P. Dale_, Dec 04 2019 *)

%t Rest[Table[With[{s=Floor[Sqrt[p]]},Abs[p-Nearest[Range[s-2,s+2]^2,p]]],{p,Prime[ Range[ 100]]}]//Flatten] (* _Harvey P. Dale_, Apr 27 2022 *)

%o (Python)

%o from sympy import integer_nthroot, prime

%o def A047972(n):

%o     p = prime(n)

%o     a = integer_nthroot(p,2)[0]

%o     return min(p-a**2,(a+1)**2-p) # _Chai Wah Wu_, Apr 03 2021

%Y Cf. A047973.

%K easy,nonn,nice

%O 1,4

%A _Enoch Haga_, Dec 11 1999