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A007304
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Sphenic numbers: products of 3 distinct primes.
(Formerly M5207)
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187
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30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426, 429, 430, 434, 435, 438
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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EXAMPLE
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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)
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MAPLE
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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
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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A002033, A010051, A020639, A037074, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A107768, A179643, A179695.
For the following, NNS means "not necessarily strict".
A008289 counts strict partitions by sum and length.
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009
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STATUS
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approved
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