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A007304 Sphenic numbers: products of 3 distinct primes.
(Formerly M5207)
149
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

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Index to sequences related to prime signature

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

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

CROSSREFS

Cf. A006881, A046386, A046387, A067885 (product of 2, 4, 5 and 6 distinct primes, resp.)

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".

A014612 is the NNS version.

A046389 is the restriction to odds (NNS: A046316).

A075819 is the restriction to evens (NNS: A075818).

A239656 gives first differences.

A285508 lists terms of A014612 that are not squarefree.

A307534 is the case where all prime indices are odd (NNS: A338471).

A337453 is a different ranking of ordered triples (NNS: A014311).

A338557 is the case where all prime indices are even (NNS: A338556).

A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).

A005117 lists squarefree numbers.

A008289 counts strict partitions by sum and length.

A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).

Cf. A000212, A000217, A001840, A101271, A284825, A321773, A337599, A337605.

Sequence in context: A225228 A336568 A093599 * A160350 A053858 A075819

Adjacent sequences:  A007301 A007302 A007303 * A007305 A007306 A007307

KEYWORD

nonn,easy

AUTHOR

Simon Plouffe

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 February 28 02:10 EST 2021. Contains 341695 sequences. (Running on oeis4.)