A178740 Product of the 5th power of a prime (A050997) and a different prime (p^5*q).

96, 160, 224, 352, 416, 486, 544, 608, 736, 928, 992, 1184, 1215, 1312, 1376, 1504, 1696, 1701, 1888, 1952, 2144, 2272, 2336, 2528, 2656, 2673, 2848, 3104, 3159, 3232, 3296, 3424, 3488, 3616, 4064, 4131, 4192, 4384, 4448, 4617, 4768, 4832, 5024, 5216

Offset: 1

Author

Will Nicholes, Jun 08 2010

Keywords

• easy
• nonn

Comments

Subsequence of A030630, integers whose number of divisors is 12. - Michel Marcus, Nov 11 2015

Links

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

Crossrefs

Cf. A178739, A050997 and A143610.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,5};Select[Range[6000],f] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
With[{nn=50},Take[Union[Flatten[{#[[1]]^5 #[[2]],#[[1]]#[[2]]^5}&/@Subsets[ Prime[ Range[nn]],{2}]]],nn]] (* Harvey P. Dale, Mar 18 2013 *)

PARI

list(lim)=my(v=List(), t); forprime(p=2, (lim\2)^(1/5), t=p^5; forprime(q=2, lim\t, if(p==q, next); listput(v, t*q))); vecsort(Vec(v)) \\ Altug Alkan, Nov 11 2015

PARI

isok(n)=my(f=factor(n)[, 2]); f==[5, 1]~||f==[1, 5]~
for(n=1, 1e4, if(isok(n), print1(n,", "))) \\ Altug Alkan, Nov 11 2015

Python

from sympy import primepi, primerange, integer_nthroot
def A178740(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**5) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,6)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025