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Difference between a nontrivial prime power (A025475) and the next prime.
1

%I #17 Oct 19 2022 09:07:49

%S 1,3,2,1,4,2,5,4,3,2,6,2,3,4,8,1,4,4,6,9,12,6,4,12,6,7,30,4,12,12,5,

%T 16,6,4,10,10,12,10,6,3,4,6,10,4,6,2,4,10,6,17,4,10,4,18,6,30,12,12,4,

%U 10,27,4,6,4,12,4,28,6,2,10,4,4,10,12,18,10,10,3,12,4,12,6,10,10,18,10,12

%N Difference between a nontrivial prime power (A025475) and the next prime.

%C a(n)=1 only for some powers of 2 corresponding to Fermat primes > 3. - Edited by _Robert Israel_, Jun 03 2021

%H Robert Israel, <a href="/A060848/b060848.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = nextprime(A025475(n)) - A025475(n) = A013632(A025475(n)).

%e 78125=5^7 is followed by 78137, the difference is 12.

%p N:= 10^5: # for prime powers <= N

%p S:= {}:

%p p:= 1:

%p do

%p p:= nextprime(p);

%p if p^2 > N then break fi;

%p S:= S union {seq(p^i,i=2..floor(log[p](N)))}

%p od:

%p map(t -> nextprime(t)-t, sort(convert(S,list))); # _Robert Israel_, Jun 03 2021

%t NextPrime[#]-#&/@Select[Range[100000],PrimePowerQ[#]&&!PrimeQ[#]&] (* _Harvey P. Dale_, Oct 19 2022 *)

%Y Cf. A025475, A007918, A013632.

%K nonn

%O 1,2

%A _Labos Elemer_, May 03 2001