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A110289
7-almost primes p*q*r*s*t*u*v relatively prime to p+q+r+s+t+u+v.
12
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
1,1
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
LINKS
EXAMPLE
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.
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 *)
PROG
(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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jul 18 2005
EXTENSIONS
Extended by Ray Chandler and Rick L. Shepherd, Jul 20 2005
STATUS
approved