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

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

Author

Altug Alkan, Mar 24 2018

Keywords

• nonn
• look

Examples

			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.
		

Links

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

Crossrefs

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).

Programs

Maple

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

PARI

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; }

Formula

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