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 A007304 Sphenic numbers: products of 3 distinct primes. (Formerly M5207) 116
 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 (list; graph; refs; listen; history; text; internal format)
 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 parallelopiped 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 parallelopipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007 Or the numbers n such that 13 = number of perfect partitions of n. - Juri-Stepan Gerasimov, Oct 07 2009 A178254(a(n)) = 36. - Reinhard Zumkeller, May 24 2010 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 Also numbers n with A001222(n)=3 and A001221(n)=3. - Enrique Pérez Herrero, Jun 28 2012 A050326(a(n)) = 5, subsequence of A225228; A162143(n) = a(n)^2. - Reinhard Zumkeller, May 03 2013 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. LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 FORMULA A008683(a(n)) = -1. A000005(a(n)) = 8. - R. J. Mathar, Aug 14 2009 A002033(a(n)-1) = 13. - Juri-Stepan Gerasimov, Oct 07 2009, R. J. Mathar, Oct 14 2009 a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015 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 *) 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 CROSSREFS Subsequence of A014612. Cf. A006881, A046386, A046387, A067885 (product of 2, 4, 5 and 6 distinct primes, resp.) Cf. A046389, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A037074, A107768, A002033, A179643, A179695, A020639, A010051, A239656 (first differences). Sequence in context: A238367 A225228 A093599 * A160350 A053858 A075819 Adjacent sequences:  A007301 A007302 A007303 * A007305 A007306 A007307 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 Formula index corrected - R. J. Mathar, Oct 14 2009 STATUS approved

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Last modified January 23 19:25 EST 2019. Contains 319404 sequences. (Running on oeis4.)