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A060847 Difference between a nontrivial prime power (A246547) and the previous prime.

%I A060847 #17 Nov 13 2024 16:59:03
%S A060847 1,1,2,3,2,4,1,2,3,2,8,12,1,2,2,5,6,6,2,3,6,6,2,2,8,3,4,2,12,2,9,8,18,
%T A060847 2,2,6,4,12,2,3,6,4,2,6,12,8,2,6,2,1,6,8,2,2,14,4,6,2,6,2,3,20,2,12,2,
%U A060847 2,8,14,10,18,8,6,2,2,2,12,12,19,2,6,6,20,2,2,2,8,8,2,2,8,20,12,15,2,4
%N A060847 Difference between a nontrivial prime power (A246547) and the previous prime.
%C A060847 a(n)=1 only for some powers of 2.
%H A060847 Robert Israel, <a href="/A060847/b060847.txt">Table of n, a(n) for n = 1..10000</a>
%F A060847 a(n) = A246547(n)-prevprime(A246547(n)) = A246547(n)-A049711(A246547(n)).
%e A060847 78125=5^7 follows 78121, the difference is 4.
%p A060847 N:= 10^5: # to consider prime powers <= N
%p A060847 P:= select(isprime,[2,seq(i,i=3..floor(sqrt(N)),2)]):
%p A060847 PP:= sort([seq(seq(p^k,k=2..ilog[p](N)),p=P)]):
%p A060847 map(t -> t - prevprime(t), PP); # _Robert Israel_, Nov 13 2024
%o A060847 (Python)
%o A060847 from sympy import primepi, integer_nthroot, prevprime
%o A060847 def A060847(n):
%o A060847     def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
%o A060847     def bisection(f,kmin=0,kmax=1):
%o A060847         while f(kmax) > kmax: kmax <<= 1
%o A060847         while kmax-kmin > 1:
%o A060847             kmid = kmax+kmin>>1
%o A060847             if f(kmid) <= kmid:
%o A060847                 kmax = kmid
%o A060847             else:
%o A060847                 kmin = kmid
%o A060847         return kmax
%o A060847     return (a:=bisection(f,n,n))-prevprime(a) # _Chai Wah Wu_, Sep 13 2024
%Y A060847 Cf. A025475, A000961, A001597, A001694, A007917, A007918, A013632, A013633, A049711, A246547.
%K A060847 nonn
%O A060847 1,3
%A A060847 _Labos Elemer_, May 03 2001