A178739 Product of the 4th power of a prime (A030514) and a different prime (p^4*q).

48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 567, 592, 656, 688, 752, 848, 891, 944, 976, 1053, 1072, 1136, 1168, 1250, 1264, 1328, 1377, 1424, 1539, 1552, 1616, 1648, 1712, 1744, 1808, 1863, 1875, 2032, 2096, 2192, 2224, 2349, 2384, 2416, 2511

Offset: 1

Author

Will Nicholes, Jun 08 2010

Keywords

• easy
• nonn

Comments

Subsequence of A030628.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index to sequences related to prime signature

Crossrefs

Cf. A065036, A030514, A030628, A085986, A085987.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,4}; Select[Range[10000], f] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
max = 500000; A178739 = DeleteCases[Union[Table[Prime[p] Prime[q]^4 Boole[p != q], {p, PrimePi[max/16]}, {q, PrimePi[max/2]}]], 0]; Take[A178739, 50] (* Alonso del Arte, Aug 05 2012 *)

PARI

list(lim)=my(v=List(),t);forprime(p=2,(lim\2)^(1/4), t=p^4; forprime(q=2,lim\t, if(p==q,next); listput(v,t*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Python

from sympy import primepi, primerange, integer_nthroot
def A178739(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(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,5)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Formula

a(n) ~ kn log n with k = 1/P(4) = 1/A085964 = 12.98817.... - Charles R Greathouse IV, Feb 23 2017