A062780 - OEIS

%I #8 Sep 12 2024 11:23:14

%S 4,1,16,2,5,17,72,4,3,41,74,46,54,18,168,312,120,370,38,312,168,199,

%T 139,10,12,600,316,356,240,768,424,128,288,912,618,30,1032,271,1217,

%U 792,408,840,432,286,602,3360,678,354,1608,552,2880,600,1588,260,1920,1320,1902

%N Differences between consecutive prime powers of primes (see A053810).

%F a(n) = A053810(n+1) - A053810(n).

%e 11^2=121 and 5^3=125 are members with index 8 and 9 in A053810. So a(8)=125-121=4.

%o (Python)

%o from sympy import primepi, integer_nthroot, primerange

%o def A062780(n):

%o     def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     return -(a:=bisection(f,n,n))+bisection(lambda x:f(x)+1,a,a) # _Chai Wah Wu_, Sep 12 2024

%Y Cf. A000961, A058310, A057820.

%K easy,nonn,changed

%O 1,1

%A _Rainer Rosenthal_, Jul 18 2001

%E Edited and extended by _Ray Chandler_, Oct 30 2008