A113877 Semiprimes to semiprime powers.

256, 1296, 4096, 6561, 10000, 38416, 46656, 50625, 194481, 234256, 262144, 390625, 456976, 531441, 1000000, 1048576, 1185921, 1336336, 1500625, 2085136, 2313441, 4477456, 5764801, 6765201, 7529536, 9150625, 10077696, 10556001, 11316496, 11390625, 14776336

Offset: 1

Author

Jonathan Vos Post, Jan 27 2006

Keywords

• easy
• nonn

Comments

This is the semiprime analog of A053810.

Examples

			a(1) = 256 = 4^4 = semiprime(1)^semiprime(1).
a(2) = 1296 = 6^4 = semiprime(2)^semiprime(1).
a(3) = 4096 = 4^6 = semiprime(1)^semiprime(2).
a(4) = 6561 = 9^4 = semiprime(3)^semiprime(1).
a(5) = 10000 = 10^4.
		

Links

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

Crossrefs

Cf. A001358, A113854, A053810.

Programs

Mathematica

lim = 10^8; s = Select[Range[lim^(1/4)], Total[Transpose[FactorInteger[#]][[2]]] == 2 &]; t = {}; j = 1; While[b = s[[j]]; i = 1; While[a = s[[i]]; e = a^b; If[e <= lim, AppendTo[t, e]]; e < lim && i < Length[s], i++]; i > 1, j++]; t = Union[t] (* T. D. Noe, Jun 05 2013 *)

PARI

is(n)=my(b,e=ispower(n,,&b),o); if(e==0,return(0)); o=bigomega(e); (o==2 && bigomega(b)==2) || (e%2==0 && o==3 && isprime(b)) \\ Charles R Greathouse IV, Jun 05 2013

PARI

list(lim)=my(v=List());for(e=4,log(lim\=1+.5)\log(4), if(bigomega(e)!=2, next); for(b=4,(lim+.5)^(1/e), if(bigomega(b)==2, listput(v,b^e)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 05 2013

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, factorint
def A113877(n):
    def A072000(n): return int(-((t:=primepi(s:=isqrt(n)))*(t-1)>>1)+sum(primepi(n//p) for p in primerange(s+1)))
    def f(x): return int(n+x-sum(A072000(integer_nthroot(x, p)[0]) for p in range(4,x.bit_length()) if sum(factorint(p).values())==2))
    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 bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

Formula

{a(n)} = {a^b where a and b are elements of A001358}.

{a(n)} = {(p*q)^(r*s) = (p^(r*s))*(q^r*s) for distinct primes p, q, r, s} UNION {(p*q)^(p*r) = (p^(p*r))*(q^(p*r)) for distinct primes p, q, r} UNION {(p*q)^(r*r) = (p^(r^2))*(q^(r^2)) for distinct primes p, q, r} UNION {(p*q)^(p*q)= (p^(p*q))*(q^(p*q)) for distinct primes p, q} UNION {(p^2)^(p^2) = p^(2*(p^2)) for prime p}.

a(n) ~ (n log n/log log n)^4. - Charles R Greathouse IV, Jun 05 2013

Extensions

Terms corrected by Charles R Greathouse IV, Jun 05 2013