OFFSET
1,1
COMMENTS
Note the distinctions between this and "n has exactly three prime factors" (A014612) or "n has exactly three distinct prime factors." (A033992). The word "sphenic" also means "shaped like a wedge" [American Heritage Dictionary] as in dentation with "sphenic molars." - Jonathan Vos Post, Sep 11 2005
Also the volume of a sphenic brick. A sphenic brick is a rectangular parallelepiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. Example: The distinct prime triple (3,5,7) produces a 3x5x7 unit brick which has volume 105 cubic units. 3-D analog of 2-D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3-D A107768 Golden 3-almost primes = Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007
Sum(n>=1, 1/a(n)^s) = (1/6)*(P(s)^3 - P(3*s) - 3*(P(s)*P(2*s)-P(3*s))), where P is prime zeta function. - Enrique Pérez Herrero, Jun 28 2012
n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - Peter Dolland, Apr 11 2020
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
"Sphenic", The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Mifflin Company, 2000.
FORMULA
A008683(a(n)) = -1.
A000005(a(n)) = 8. - R. J. Mathar, Aug 14 2009
A178254(a(n)) = 36. - Reinhard Zumkeller, May 24 2010
a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015
EXAMPLE
From Gus Wiseman, Nov 05 2020: (Start)
Also Heinz numbers of strict integer partitions into three parts, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are counted by A001399(n-6) = A069905(n-3), with ordered version A001399(n-6)*6. The sequence of terms together with their prime indices begins:
30: {1,2,3} 182: {1,4,6} 286: {1,5,6}
42: {1,2,4} 186: {1,2,11} 290: {1,3,10}
66: {1,2,5} 190: {1,3,8} 310: {1,3,11}
70: {1,3,4} 195: {2,3,6} 318: {1,2,16}
78: {1,2,6} 222: {1,2,12} 322: {1,4,9}
102: {1,2,7} 230: {1,3,9} 345: {2,3,9}
105: {2,3,4} 231: {2,4,5} 354: {1,2,17}
110: {1,3,5} 238: {1,4,7} 357: {2,4,7}
114: {1,2,8} 246: {1,2,13} 366: {1,2,18}
130: {1,3,6} 255: {2,3,7} 370: {1,3,12}
138: {1,2,9} 258: {1,2,14} 374: {1,5,7}
154: {1,4,5} 266: {1,4,8} 385: {3,4,5}
165: {2,3,5} 273: {2,4,6} 399: {2,4,8}
170: {1,3,7} 282: {1,2,15} 402: {1,2,19}
174: {1,2,10} 285: {2,3,8} 406: {1,4,10}
(End)
MAPLE
with(numtheory): a:=proc(n) if bigomega(n)=3 and nops(factorset(n))=3 then n else fi end: seq(a(n), n=1..450); # Emeric Deutsch
A007304 := proc(n)
option remember;
local a;
if n =1 then
30;
else
for a from procname(n-1)+1 do
if bigomega(a)=3 and nops(factorset(a))=3 then
return a;
end if;
end do:
end if;
end proc: # R. J. Mathar, Dec 06 2016
MATHEMATICA
Union[Flatten[Table[Prime[n]*Prime[m]*Prime[k], {k, 20}, {n, k+1, 20}, {m, n+1, 20}]]]
Take[ Sort@ Flatten@ Table[ Prime@i Prime@j Prime@k, {i, 3, 21}, {j, 2, i - 1}, {k, j - 1}], 53] (* Robert G. Wilson v *)
With[{upto=500}, Sort[Select[Times@@@Subsets[Prime[Range[Ceiling[upto/6]]], {3}], #<=upto&]]] (* Harvey P. Dale, Jan 08 2015 *)
Select[Range[100], SquareFreeQ[#]&&PrimeOmega[#]==3&] (* Gus Wiseman, Nov 05 2020 *)
PROG
(PARI) for(n=1, 1e4, if(bigomega(n)==3 && omega(n)==3, print1(n", "))) \\ Charles R Greathouse IV, Jun 10 2011
(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim)^(1/3), forprime(q=p+1, sqrt(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
(Haskell)
a007304 n = a007304_list !! (n-1)
a007304_list = filter f [1..] where
f u = p < q && q < w && a010051 w == 1 where
p = a020639 u; v = div u p; q = a020639 v; w = div v q
-- Reinhard Zumkeller, Mar 23 2014
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A007304(n):
def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a, k in enumerate(primerange(integer_nthroot(x, 3)[0]+1), 1) for b, m in enumerate(primerange(k+1, isqrt(x//k)+1), a+1)))
kmin, kmax = 0, 1
while f(kmax) > kmax:
kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax # Chai Wah Wu, Aug 29 2024
CROSSREFS
Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A002033, A010051, A020639, A037074, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A107768, A179643, A179695.
Cf. A162143 (a(n)^2).
For the following, NNS means "not necessarily strict".
A014612 is the NNS version.
A239656 gives first differences.
A005117 lists squarefree numbers.
A008289 counts strict partitions by sum and length.
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Jan 04 2006
Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009
STATUS
approved