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
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
Programs
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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
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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
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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
Comments