A062780 Differences between consecutive prime powers of primes (see A053810).

4, 1, 16, 2, 5, 17, 72, 4, 3, 41, 74, 46, 54, 18, 168, 312, 120, 370, 38, 312, 168, 199, 139, 10, 12, 600, 316, 356, 240, 768, 424, 128, 288, 912, 618, 30, 1032, 271, 1217, 792, 408, 840, 432, 286, 602, 3360, 678, 354, 1608, 552, 2880, 600, 1588, 260, 1920, 1320, 1902

Offset

1,1

Links

Table of n, a(n) for n=1..57.

Formula

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

Example

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

Prog

(Python)
from sympy import primepi, integer_nthroot, primerange
def A062780(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return -(a:=bisection(f, n, n))+bisection(lambda x:f(x)+1, a, a) # Chai Wah Wu, Sep 12 2024

Crossrefs

Cf. A000961, A058310, A057820.

Sequence in context: A146160 A059222 A117292 * A262616 A309074 A175844

Adjacent sequences: A062777 A062778 A062779 * A062781 A062782 A062783

Keyword

easy,nonn

Author

Rainer Rosenthal, Jul 18 2001

Extensions

Edited and extended by Ray Chandler, Oct 30 2008

Status

approved