A061670 Distance to nearest prime power p^k, k=0 and k >= 2 (A025475).

0, 1, 1, 0, 1, 2, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 1

Author

Michel ten Voorde, Jun 16 2001

Keywords

• nonn
• look

Examples

			a(12)=3 because 9=3^2 is the nearest power to 12 (12-9=3).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to distance to nearest element of some set

Crossrefs

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

N:= 1000: # to get a(1)..a(M) where M is the greatest prime power <= N.
Primes:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)))]):
Pows:= sort(convert({1,seq(seq(p^e,e=2..floor(log[p](N))),p=Primes)},list)):
nP:= nops(Pows):
M:= Pows[nP]:
V:= Vector(M):
V[2]:= 1:
for i from 2 to nP-1 do
  for x from ceil((Pows[i]+Pows[i-1])/2) to floor((Pows[i]+Pows[i+1])/2) do
    V[x]:= abs(x - Pows[i])
od od:
for x from ceil((M+Pows[nP-1])/2) to M do V[x]:= M - x od:
convert(V,list); # Robert Israel, Mar 23 2018

PARI

isA025475(n) = {isprimepower(n) && !isprime(n) || n==1}
a(n) = {my(k=0); while(!isA025475(n+k) && !isA025475(n-k), k++); k; } \\ Altug Alkan, Mar 23 2018

Extensions

Definition corrected, and more terms from Robert Israel, Mar 23 2018