A047973 -id:A047973 - cp's OEIS frontend

A047972 Distance of n-th prime to nearest square.

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, 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, 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

Offset: 1

Enoch Haga, Dec 11 1999

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

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

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A047973.

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 *)
Table[Apply[Min, Abs[p - Through[{Floor, Ceiling}[Sqrt[p]]]^2]], {p, Prime[Range[90]]}] (* Jan Mangaldan, May 07 2014 *)
Min[Abs[#-Through[{Floor,Ceiling}[Sqrt[#]]]^2]]&/@Prime[Range[100]] (* More concise version of program immediately above *) (* Harvey P. Dale, Dec 04 2019 *)
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 *)

from sympy import integer_nthroot, prime
def A047972(n):
    p = prime(n)
    a = integer_nthroot(p,2)[0]
    return min(p-a**2,(a+1)**2-p) # Chai Wah Wu, Apr 03 2021

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

A301630 a(n) = distance of n-th prime to nearest prime power p^k, k=0 and k >= 2 (A025475).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 2, 2, 1, 5, 8, 6, 2, 4, 5, 3, 3, 7, 8, 2, 2, 8, 16, 20, 18, 14, 12, 8, 1, 3, 9, 11, 20, 18, 12, 6, 2, 4, 10, 12, 22, 24, 28, 30, 32, 20, 16, 14, 10, 4, 2, 5, 1, 7, 13, 15, 12, 8, 6, 4, 18, 22, 24, 26, 12, 6, 4, 6, 8, 2, 6, 12, 18, 22, 28, 36, 40, 48, 58, 60, 70, 72, 73, 69, 63, 55

Offset: 1

Altug Alkan, Mar 24 2018

			a(9) = a(10) = 2 because 5^2 is the nearest prime power (A025475) to prime(9) = 23 and 3^3 is the nearest prime power (A025475) to prime(10) = 29.

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A047972, A047973, A061670.

There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2).

Primes:= select(isprime, [2,seq(i,i=3..1000,2)]):
Ppows:= sort([1,seq(seq(p^j, j=2..floor(log[p](1000))),p=Primes)]):
for n from 1 while Primes[n] < Ppows[-1] do
  i:= ListTools:-BinaryPlace(Ppows,Primes[n]);
  A[n]:= min(Primes[n]-Ppows[i],Ppows[i+1]-Primes[n])
od:
seq(A[i],i=1..n-1); # Robert Israel, Mar 26 2018

isA025475(n) = {isprimepower(n) && !isprime(n) || n==1}
a(n) = {my(k=1, p=prime(n)); while(!isA025475(p+k) && !isA025475(p-k), k++); k; }

a(n) = A061670(A000040(n)).

cp's OEIS Frontend

A047972 Distance of n-th prime to nearest square.

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