%I M5207 #107 Aug 30 2024 10:17:06

%S 30,42,66,70,78,102,105,110,114,130,138,154,165,170,174,182,186,190,

%T 195,222,230,231,238,246,255,258,266,273,282,285,286,290,310,318,322,

%U 345,354,357,366,370,374,385,399,402,406,410,418,426,429,430,434,435,438

%N Sphenic numbers: products of 3 distinct primes.

%C 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

%C 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

%C 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

%C Also numbers n with A001222(n)=3 and A001221(n)=3. - _Enrique Pérez Herrero_, Jun 28 2012

%C n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - _Peter Dolland_, Apr 11 2020

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D "Sphenic", The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Mifflin Company, 2000.

%H T. D. Noe, <a href="/A007304/b007304.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>

%F A008683(a(n)) = -1.

%F A000005(a(n)) = 8. - _R. J. Mathar_, Aug 14 2009

%F A002033(a(n)-1) = 13. - _Juri-Stepan Gerasimov_, Oct 07 2009, _R. J. Mathar_, Oct 14 2009

%F A178254(a(n)) = 36. - _Reinhard Zumkeller_, May 24 2010

%F A050326(a(n)) = 5, subsequence of A225228. - _Reinhard Zumkeller_, May 03 2013

%F a(n) ~ 2n log n/(log log n)^2. - _Charles R Greathouse IV_, Sep 14 2015

%e From _Gus Wiseman_, Nov 05 2020: (Start)

%e 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:

%e 30: {1,2,3} 182: {1,4,6} 286: {1,5,6}

%e 42: {1,2,4} 186: {1,2,11} 290: {1,3,10}

%e 66: {1,2,5} 190: {1,3,8} 310: {1,3,11}

%e 70: {1,3,4} 195: {2,3,6} 318: {1,2,16}

%e 78: {1,2,6} 222: {1,2,12} 322: {1,4,9}

%e 102: {1,2,7} 230: {1,3,9} 345: {2,3,9}

%e 105: {2,3,4} 231: {2,4,5} 354: {1,2,17}

%e 110: {1,3,5} 238: {1,4,7} 357: {2,4,7}

%e 114: {1,2,8} 246: {1,2,13} 366: {1,2,18}

%e 130: {1,3,6} 255: {2,3,7} 370: {1,3,12}

%e 138: {1,2,9} 258: {1,2,14} 374: {1,5,7}

%e 154: {1,4,5} 266: {1,4,8} 385: {3,4,5}

%e 165: {2,3,5} 273: {2,4,6} 399: {2,4,8}

%e 170: {1,3,7} 282: {1,2,15} 402: {1,2,19}

%e 174: {1,2,10} 285: {2,3,8} 406: {1,4,10}

%e (End)

%p 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_

%p A007304 := proc(n)

%p option remember;

%p local a;

%p if n =1 then

%p 30;

%p else

%p for a from procname(n-1)+1 do

%p if bigomega(a)=3 and nops(factorset(a))=3 then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc: # _R. J. Mathar_, Dec 06 2016

%t Union[Flatten[Table[Prime[n]*Prime[m]*Prime[k], {k, 20}, {n, k+1, 20}, {m, n+1, 20}]]]

%t 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_ *)

%t With[{upto=500},Sort[Select[Times@@@Subsets[Prime[Range[Ceiling[upto/6]]],{3}],#<=upto&]]] (* _Harvey P. Dale_, Jan 08 2015 *)

%t Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==3&] (* _Gus Wiseman_, Nov 05 2020 *)

%o (PARI) for(n=1,1e4,if(bigomega(n)==3 && omega(n)==3,print1(n", "))) \\ _Charles R Greathouse IV_, Jun 10 2011

%o (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

%o (Haskell)

%o a007304 n = a007304_list !! (n-1)

%o a007304_list = filter f [1..] where

%o f u = p < q && q < w && a010051 w == 1 where

%o p = a020639 u; v = div u p; q = a020639 v; w = div v q

%o -- _Reinhard Zumkeller_, Mar 23 2014

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot

%o def A007304(n):

%o 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)))

%o kmin, kmax = 0,1

%o while f(kmax) > kmax:

%o kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax # _Chai Wah Wu_, Aug 29 2024

%Y Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.

%Y Cf. A002033, A010051, A020639, A037074, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A107768, A179643, A179695.

%Y Cf. A162143 (a(n)^2).

%Y For the following, NNS means "not necessarily strict".

%Y A014612 is the NNS version.

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

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

%Y A239656 gives first differences.

%Y A285508 lists terms of A014612 that are not squarefree.

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

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

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

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

%Y A005117 lists squarefree numbers.

%Y A008289 counts strict partitions by sum and length.

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

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

%K nonn,easy

%O 1,1

%A _Simon Plouffe_

%E More terms from _Robert G. Wilson v_, Jan 04 2006

%E Comment concerning number of divisors corrected by _R. J. Mathar_, Aug 14 2009