A110289 7-almost primes p*q*r*s*t*u*v relatively prime to p+q+r+s+t+u+v.

320, 432, 448, 704, 720, 832, 972, 1088, 1216, 1472, 1584, 1680, 1856, 1984, 2000, 2268, 2352, 2368, 2448, 2624, 2700, 2752, 3008, 3120, 3312, 3392, 3645, 3696, 3776, 3904, 3920, 4176, 4212, 4288, 4368, 4400, 4544, 4672, 5056, 5103, 5200, 5312, 5488

Offset: 1

Author

Jonathan Vos Post, Jul 18 2005

Keywords

• easy
• nonn

Comments

The primes p, q, r, s, t, u, v are not necessarily distinct. The 7-almost primes are A046308. The converse, A110290, is 7-almost primes p*q*r*s*t*u*v which are not relatively prime to p+q+r+s+t+u+v.

Contains p*q^6 if p and q are distinct primes, p >= 5. - Robert Israel, Jan 13 2017

Examples

			832 = 2^6 * 13 is in this sequence because its sum of prime factors is 2 + 2 + 2 + 2 + 2 + 2 + 13 = 25 = 5^2, which has no factor in common with 832.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046308, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110290, A110296, A110297.

Cf. A001414 (sopfr(n)).

Programs

Maple

N:= 10^4: # to get all terms <= N
P:= select(isprime, [$1..N/2^6]):
nP:= nops(P):
Res:= {}:
for p in P do
  for q in P while q <= p and p*q*2^5 <= N do
    for r in P while r <= q and p*q*r*2^4 <= N do
      for s in P while s <= r and p*q*r*s*2^3 <= N do
        for t in P while t <= s and p*q*r*s*t*2^2 <= N do
          for u in P while u <= t and p*q*r*s*t*u*2 <= N do
            for v in P while v <= u and p*q*r*s*t*u*v <= N do
              if igcd(p+q+r+s+t+u+v,p*q*r*s*t*u*v) = 1 then
                  Res:= Res union {p*q*r*s*t*u*v} fi
od od od od od od od:
sort(convert(Res,list)); # Robert Israel, Jan 13 2017

Mathematica

Select[Range[6000],PrimeOmega[#]==7&&CoprimeQ[Total[ Times@@@ FactorInteger[ #]],#]&] (* Harvey P. Dale, Nov 19 2019 *)

PARI

sopfr(n)=local(f);if(n<1,0,f=factor(n);sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))
isok(n)=bigomega(n)==7&&gcd(n, sopfr(n))==1 \\ Rick L. Shepherd, Jul 20 2005

Extensions

Extended by Ray Chandler and Rick L. Shepherd, Jul 20 2005