A181970 Places of nonprimes in A050376.

3, 6, 9, 13, 20, 28, 37, 47, 63, 71, 83, 111, 127, 160, 177, 235, 280, 301, 348, 377, 430, 509, 542, 633, 700, 731, 838, 875, 915, 1030, 1194, 1284, 1327, 1415, 1458, 1559, 1752, 1915, 2015, 2181, 2231, 2531, 2590, 2773, 2960, 3089, 3154, 3289, 3485, 3562, 3919, 3997, 4142

Offset: 1

Author

Vladimir Shevelev, Apr 06 2012

Keywords

• nonn

Comments

Or numbers not expressed in the form pi(p) + pi(sqrt(p)) + pi((sqrt(sqrt(p)))) +... with prime p.

Examples

			Show that 28 not expressed in form pi(p) + pi(sqrt(p)) + pi((sqrt(sqrt(p)))) +... with prime p. Indeed, for p=79, this sum is 22+4+1=27, while for p=83, it is 23+4+2=29.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A050376, A000040, A181962.

Programs

PARI

first_few(lim)=my(v=List(apply(n->n^2, primes(primepi(sqrtint(lim))))),u,t); forprime(p=2,(lim+.5)^(1/4),t=p^2;while((t=t^2)<=lim,listput(v,t)));listput(v,1);v=vecsort(Vec(v));u=vector(#v-1,i,sum(j=v[i]+1,v[i+1]-1,isprime(j)));u[1]++;for(i=2, #u, u[i]+=u[i-1]+1);u \\ Charles R Greathouse IV, Apr 10 2012

Python

from sympy import primepi, integer_nthroot
def A181970(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(integer_nthroot(x,1<Chai Wah Wu, Feb 18-19 2025

Formula

a(n) ~ 0.5 n^2 log n. - Charles R Greathouse IV, Apr 11 2012

Extensions

a(30)-a(53) from Charles R Greathouse IV, Apr 10 2012