A053707 First differences of A025475, powers of a prime but not prime.

3, 4, 1, 7, 9, 2, 5, 17, 15, 17, 40, 4, 3, 41, 74, 13, 33, 54, 18, 151, 17, 96, 104, 112, 120, 63, 307, 38, 312, 168, 199, 139, 10, 12, 192, 408, 316, 356, 240, 375, 393, 424, 128, 288, 912, 320, 298, 30, 1032, 271, 1217, 792, 408, 840, 432, 286, 602, 1872, 984, 504

Offset: 1

Author

Labos Elemer, Feb 14 2000

Keywords

• nonn

Comments

In the graph of this sequence, the lowest curve corresponds to differences of squares of twin primes; the next-lowest curve is for squares of adjacent primes differing by 4, etc. - T. D. Noe, Aug 03 2007

Examples

			2^0 = 1 is the first number that meets the definition of A025475, the next one is 2^2 = 4, hence a(1) = 4 - 1 = 3.
a(3) = A025475(4) - A025475(3) = 9 - 8 = 1; a(11) = A025475(12) - A025475(11) = 121 - 81 = 40.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000

Crossrefs

Cf. A025475.

Programs

Mathematica

Differences@ Join[{1}, Select[Range@ 16200, And[! PrimeQ@ #, PrimePowerQ@ #] &]] (* Michael De Vlieger, Jul 04 2016 *)

PARI

{k=1; for(n=2,16300,if(matsize(factor(n))[1]==1&&factor(n)[1,2]>1,d=n-k; print1(d,","); k=n))} \\ Klaus Brockhaus, Sep 25 2003

Python

from sympy import primepi, integer_nthroot
def A053707(n):
    if n==1: return 3
    def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax)+1 >= kmax:
        kmax <<= 1
    rmin, rmax = 1, kmax
    while True:
        kmid = kmax+kmin>>1
        if f(kmid)+1 < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    while True:
        rmid = rmax+rmin>>1
        if f(rmid) < rmid:
            rmax = rmid
        else:
            rmin = rmid
        if rmax-rmin <= 1:
            break
    return kmax-rmax # Chai Wah Wu, Aug 13 2024

Formula

a(n) = A025475(n+1) - A025475(n).

Extensions

Edited by Klaus Brockhaus, Sep 25 2003