A179669 Products of form p^4*q^2*r where p, q and r are three distinct primes.

720, 1008, 1200, 1584, 1620, 1872, 2268, 2352, 2448, 2736, 2800, 3312, 3564, 3920, 4050, 4176, 4212, 4400, 4464, 5200, 5328, 5508, 5808, 5904, 6156, 6192, 6768, 6800, 7452, 7500, 7600, 7632, 7938, 8112, 8496, 8624, 8784, 9200, 9396, 9648, 9680, 10044

Offset

1,1

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Will Nicholes, List of Prime Signatures

Index to sequences related to prime signature

Maple

N:= 20000: # for terms < N
P:= select(isprime, [2, seq(i, i=1..N/(3^2*2^4), 2)]):
R:= NULL:
for i from 1 while P[i]^4 * 2^2*3 < N do
  for j from 1 while P[i]^4 * P[j]^2 *2 < N do
    if j = i then next fi;
    m:= ListTools:-BinaryPlace(P, N/P[i]^4/P[j]^2);
    R:= R, seq(P[i]^4*P[j]^2*P[k], k={$1..m} minus {i, j});
od od:
sort([R]); # Robert Israel, Mar 28 2025

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1, 2, 4}; Select[Range[10000], f]

Prog

(PARI) list(lim)=my(v=List(), t1, t2); forprime(p=2, (lim\12)^(1/4), t1=p^4; forprime(q=2, sqrt(lim\t1), if(p==q, next); t2=t1*q^2; forprime(r=2, lim\t2, if(p==r||q==r, next); listput(v, t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179669(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*q**2)) for p in primerange(integer_nthroot(x, 4)[0]+1) for q in primerange(isqrt(x//p**4)+1))+sum(primepi(integer_nthroot(x//p**4, 3)[0]) for p in primerange(integer_nthroot(x, 4)[0]+1))+sum(primepi(isqrt(x//p**5)) for p in primerange(integer_nthroot(x, 5)[0]+1))+sum(primepi(x//p**6) for p in primerange(integer_nthroot(x, 6)[0]+1))-(primepi(integer_nthroot(x, 7)[0])<<1)
    return bisection(f, n, n) # Chai Wah Wu, Mar 28 2025

Crossrefs

Cf. A137493.

Sequence in context: A337036 A257416 A137493 * A067808 A302127 A291804

Adjacent sequences: A179666 A179667 A179668 * A179670 A179671 A179672

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

Status

approved