A321773 -id:A321773 - cp's OEIS frontend

A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244

Offset: 1

Eric W. Weisstein

nonn

Sometimes called "triprimes" or "3-almost primes".
See also A001358 for product of two primes (sometimes called semiprimes).
If you graph a(n)/n for n up to 10000 (and probably quite a bit higher), it appears to be converging to something near 3.9. In fact the limit is infinite. - Franklin T. Adams-Watters, Sep 20 2006
Meng shows that for any sufficiently large odd integer n, the equation n = a + b + c has solutions where each of a, b, c is 3-almost prime. The number of such solutions is (log log n)^6/(16 (log n)^3)*n^2*s(n)*(1 + O(1/log log n)), where s(n) = Sum_{q >= 1} Sum_{a = 1..q, (a, q) = 1} exp(i*2*Pi*n*a/q)*mu(n)/phi(n)^3 > 1/2. - Jonathan Vos Post, Sep 16 2005, corrected & rewritten by M. F. Hasler, Apr 24 2019
Also, a(n) are the numbers such that exactly half of their divisors are composite. For the numbers in which exactly half of the divisors are prime, see A167171. - Ivan Neretin, Jan 12 2016

			From _Gus Wiseman_, Nov 04 2020: (Start)
Also Heinz numbers of integer partitions into three parts, counted by A001399(n-3) = A069905(n) with ordered version A000217, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence of terms together with their prime indices begins:
      8: {1,1,1}     70: {1,3,4}     130: {1,3,6}
     12: {1,1,2}     75: {2,3,3}     138: {1,2,9}
     18: {1,2,2}     76: {1,1,8}     147: {2,4,4}
     20: {1,1,3}     78: {1,2,6}     148: {1,1,12}
     27: {2,2,2}     92: {1,1,9}     153: {2,2,7}
     28: {1,1,4}     98: {1,4,4}     154: {1,4,5}
     30: {1,2,3}     99: {2,2,5}     164: {1,1,13}
     42: {1,2,4}    102: {1,2,7}     165: {2,3,5}
     44: {1,1,5}    105: {2,3,4}     170: {1,3,7}
     45: {2,2,3}    110: {1,3,5}     171: {2,2,8}
     50: {1,3,3}    114: {1,2,8}     172: {1,1,14}
     52: {1,1,6}    116: {1,1,10}    174: {1,2,10}
     63: {2,2,4}    117: {2,2,6}     175: {3,3,4}
     66: {1,2,5}    124: {1,1,11}    182: {1,4,6}
     68: {1,1,7}    125: {3,3,3}     186: {1,2,11}
(End)

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974). See p. 211.

T. D. Noe, Table of n, a(n) for n = 1..10000
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211.
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.
Eric Weisstein's World of Mathematics, Almost Prime

Cf. A000040, A001358 (biprimes), A014613 (quadruprimes), A033942, A086062, A098238, A123072, A123073, A101605 (characteristic function).
Cf. A109251 (number of 3-almost primes <= 10^n).
Subsequence of A145784. - Reinhard Zumkeller, Oct 19 2008
Cf. A007304 is the squarefree case.
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), this sequence (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
Cf. A253721 (final digits).
A014311 is a different ranking of ordered triples, with strict case A337453.
A046316 is the restriction to odds, with strict case A307534.
A075818 is the restriction to evens, with strict case A075819.
A285508 is the nonsquarefree case.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
Cf. A000212, A000217, A046389, A140106, A307719, A321773.

a014612 n = a014612_list !! (n-1)
a014612_list = filter ((== 3) . a001222) [1..] -- Reinhard Zumkeller, Apr 02 2012

with(numtheory); A014612:=n->`if`(bigomega(n)=3, n, NULL); seq(A014612(n), n=1..250) # Wesley Ivan Hurt, Feb 05 2014

threeAlmostPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@244, threeAlmostPrimeQ[ # ] &] (* Robert G. Wilson v, Jan 04 2006 *)
NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; NestList[NextkAlmostPrime[#, 3] &, 2^3, 56] (* Robert G. Wilson v, Jan 27 2013 *)
Select[Range[244], PrimeOmega[#] == 3 &] (* Jayanta Basu, Jul 01 2013 *)

isA014612(n)=bigomega(n)==3 \\ Charles R Greathouse IV, May 07 2011

list(lim)=my(v=List(),t);forprime(p=2,lim\4, forprime(q=2,min(lim\(2*p),p), t=p*q; forprime(r=2,min(lim\t,q),listput(v,t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 04 2013

from sympy import factorint
def ok(n): f = factorint(n); return sum(f[p] for p in f) == 3
print(list(filter(ok, range(245)))) # Michael S. Branicky, Aug 12 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A014612(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)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024

def primeFactors(number: Int, list: List[Int] = List())
                                                      : List[Int] = {
  for (n <- 2 to number if (number % n == 0)) {
    return primeFactors(number / n, list :+ n)
  }
  list
}
(1 to 250).filter(primeFactors().size == 3) // _Alonso del Arte, Nov 04 2020, based on algorithm by Victor Farcic (vfarcic)

Product p_i^e_i with Sum e_i = 3.
a(n) ~ 2n log n / (log log n)^2 as n -> infinity [Landau, p. 211].
Tau(a(n)) = 2 * (omega(a(n)) + 1) = 2*A083399(a(n)), where tau = A000005 and omega = A001221. - Wesley Ivan Hurt, Jun 28 2013
a(n) = A078840(3,n). - R. J. Mathar, Jan 30 2019

More terms from Patrick De Geest, Jun 15 1998

A007304 Sphenic numbers: products of 3 distinct primes.

Original entry on oeis.org

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

Offset: 1

Simon Plouffe

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
Also numbers n with A001222(n)=3 and A001221(n)=3. - Enrique Pérez Herrero, Jun 28 2012
n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - Peter Dolland, Apr 11 2020

			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)

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.

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.
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.

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

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
is_a := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A007304List := upto -> select(is_a, [seq(1..upto)]):  # Peter Luschny, Apr 14 2025

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 *)

for(n=1,1e4,if(bigomega(n)==3 && omega(n)==3,print1(n", "))) \\ Charles R Greathouse IV, Jun 10 2011

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

list(lim)=my(v=List(), t); forprime(p=2, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); Set(v) \\ Charles R Greathouse IV, Jan 21 2025

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

def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

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
A178254(a(n)) = 36. - Reinhard Zumkeller, May 24 2010
A050326(a(n)) = 5, subsequence of A225228. - Reinhard Zumkeller, May 03 2013
a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015

More terms from Robert G. Wilson v, Jan 04 2006

Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009

A140106 Number of noncongruent diagonals in a regular n-gon.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37

Offset: 1

Andrew McFarland, Jun 03 2008

Number of double-stars (diameter 3 trees) with n nodes. For n >= 3, number of partitions of n-2 into two parts. - Washington Bomfim, Feb 12 2011
Number of roots of the n-th Bernoulli polynomial in the left half-plane. - Michel Lagneau, Nov 08 2012
From Gus Wiseman, Oct 17 2020: (Start)
Also the number of 3-part non-strict integer partitions of n - 1. The Heinz numbers of these partitions are given by A285508. The version for partitions of any length is A047967, with Heinz numbers A013929. The a(4) = 1 through a(15) = 6 partitions are (A = 10, B = 11, C = 12):
  111  211  221  222  322  332  333  433  443  444  544  554
            311  411  331  422  441  442  533  552  553  644
                      511  611  522  622  551  633  661  662
                                711  811  722  822  733  833
                                          911  A11  922  A22
                                                    B11  C11
(End)

			The square (n=4) has two congruent diagonals; so a(4)=1. The regular pentagon also has congruent diagonals; so a(5)=1. Among all the diagonals in a regular hexagon, there are two noncongruent ones; hence a(6)=2, etc.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Washington Bomfim, Double-star corresponding to the partition [3,7]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for sequences related to trees

Cf. A000554, A007304, A007997, A013929, A022998.
Cf. A047967, A129194, A235451, A285508, A321773.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
Essentially the same as A004526.

A140106:= func< n | n eq 1 select 0 else Floor((n-2)/2) >;
[A140106(n): n in [1..80]]; // G. C. Greubel, Feb 10 2023

with(numtheory): for n from 1 to 80 do:it:=0:
y:=[fsolve(bernoulli(n,x) , x, complex)] : for m from 1 to nops(y) do : if Re(y[m])<0 then it:=it+1:else fi:od: printf(`%d, `,it):od:

a[1]=0; a[n_?OddQ] := (n-3)/2; a[n_] := n/2-1; Array[a, 100] (* Jean-François Alcover, Nov 17 2015 *)

a(n)=if(n>1,n\2-1,0) \\ Charles R Greathouse IV, Oct 16 2015

def A140106(n): return n-2>>1 if n>1 else 0 # Chai Wah Wu, Sep 18 2023

def A140106(n): return 0 if (n==1) else (n-2)//2
[A140106(n) for n in range(1,81)] # G. C. Greubel, Feb 10 2023

a(n) = floor((n-2)/2), for n > 1, otherwise 0. - Washington Bomfim, Feb 12 2011
G.f.: x^4/(1-x-x^2+x^3). - Colin Barker, Jan 31 2012
a(n) = floor(A129194(n-1)/A022998(n)), for n > 1. - Paul Curtz, Jul 23 2017
a(n) = A001399(n-3) - A001399(n-6). Compare to A007997(n) = A001399(n-3) + A001399(n-6). - Gus Wiseman, Oct 17 2020

More terms from Joseph Myers, Sep 05 2009

A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 3, 1, 0, 1, 0, 6, 4, 1, 0, 1, 1, 4, 4, 5, 1, 0, 1, 0, 9, 8, 15, 6, 1, 0, 1, 1, 9, 5, 15, 21, 7, 1, 0, 1, 0, 10, 8, 20, 6, 28, 8, 1, 0, 1, 1, 12, 12, 6, 96, 42, 36, 9, 1, 0, 1, 0, 15, 12, 30, 192, 168, 64, 45, 10, 1, 0, 1, 1, 13, 9, 20, 142, 238, 204, 93, 55, 11, 1

Offset: 0

Alois P. Heinz, May 25 2014

			T(5,1) = 1: [5].
T(5,3) = 6: [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].
T(5,4) = 4: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1].
T(5,5) = 1: [1,1,1,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 0,  1;
  0, 1, 1,  3,  1;
  0, 1, 0,  6,  4,  1;
  0, 1, 1,  4,  4,  5,  1;
  0, 1, 0,  9,  8, 15,  6,  1;
  0, 1, 1,  9,  5, 15, 21,  7,  1;
  0, 1, 0, 10,  8, 20,  6, 28,  8, 1;
  0, 1, 1, 12, 12,  6, 96, 42, 36, 9, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A057427, A059841 (for n>1), A321773, A321774, A321775, A321776, A321777, A321778, A321779, A321780.
Row sums give A242882.
T(2n,n) gives A321772.
Cf. A242896.

b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
      `if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, x^j*
       b(n-i*j, i-1,`if`(j=0, s, s union {j}))/j!), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n$2, {})):
seq(T(n), n=0..16);

b[n_, i_, s_] := b[n, i, s] = If[n==0, Total[s]!, If[i<1, 0, Expand[Sum[ If[j>0 && MemberQ[s, j], 0, x^j*b[n-i*j, i-1, If[j==0, s, s ~Union~ {j}] ]/j!], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

T(n)={Vecrev(((r,k,b,w)->if(!k||!r, if(r,0,w!*x^w), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b,1<Andrew Howroyd, Aug 31 2019

cp's OEIS Frontend

A014612 Numbers that are the product of exactly three (not necessarily distinct) primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Python

Scala

Formula

Extensions

A007304 Sphenic numbers: products of 3 distinct primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

SageMath

Formula

Extensions

A140106 Number of noncongruent diagonals in a regular n-gon.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

SageMath

Formula

Extensions

A242887 Number T(n,k) of compositions of n into parts with distinct multiplicities and with exactly k parts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI