A098238 -id:A098238 - 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

A073610 Number of primes of the form n-p where p is a prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 2, 2, 3, 0, 2, 2, 3, 2, 4, 0, 4, 2, 4, 2, 5, 0, 6, 2, 5, 0, 4, 0, 6, 2, 4, 2, 7, 0, 8, 0, 3, 2, 6, 0, 8, 2, 6, 2, 7, 0, 10, 2, 8, 0, 6, 0, 10, 2, 6, 0, 7, 0, 12, 2, 5, 2, 10, 0, 12, 0, 4, 2, 10, 0, 12, 2, 9, 2, 10, 0, 14, 0, 8, 2, 9, 0, 16, 2, 9, 0, 8, 0, 18, 2, 8, 0, 9, 0, 14, 0, 6

Offset: 1

Amarnath Murthy, Aug 05 2002

nonn

a(p) = 2 if p-2 is a prime else a(p) = 0. If n = 2p, p is a prime then a(n) is odd else a(n) is even. As p is counted only once and if q and n-q both are prime then the count is increased by 2. ( Analogous to the fact that perfect squares have odd number of divisors).
a(2k+1) = 2 if (2k-1) is prime, else a(2k+1)=0 (for any k). This sequence can be used to re-describe a couple of conjectures: the Goldbach conjecture == a(2n) > 0 for all n>=2; twin primes conjecture == for any n, there is a prime p>n s.t. a(p)>0.
Number of ordered ways of writing n as the sum of two primes.

			a(16) = 4 as there are 4 primes 3,5,11 and 13 such that 16-3,16-5,16-11and 16-13 are primes.

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

Cf. A061358, A065577, A107318, A098238

for i from 1 to 500 do a[i] := 0:j := 1:while(ithprime(j)

nn=20;a[x]:=Sum[x^i,{i,Table[Prime[n],{n,1,nn}]}];Drop[CoefficientList[a[x]^2,x],1]  (* Geoffrey Critzer, Nov 22 2012 *)

Vec(sum(i=1,100,x^prime(i),O(x^prime(101)))^2) \\ Charles R Greathouse IV, Jan 21 2015

G.f.: (Sum_{k>0} x^prime(k))^2. - Vladeta Jovovic, Mar 12 2005

Self-convolution of characteristic function of primes (A010051). - Graeme McRae, Jul 18 2006

Corrected and extended by Vladeta Jovovic and Sascha Kurz, Aug 06 2002

A121303 Triangle read by rows: T(n,k) is the number of compositions of n into k primes (i.e., ordered sequences of k primes having sum n; n>=2, k>=1).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 2, 3, 0, 2, 3, 1, 0, 2, 4, 4, 0, 3, 6, 6, 1, 1, 0, 6, 8, 5, 0, 2, 9, 13, 10, 1, 1, 2, 6, 16, 15, 6, 0, 3, 6, 22, 25, 15, 1, 0, 2, 10, 24, 36, 26, 7, 0, 4, 9, 22, 50, 45, 21, 1, 1, 0, 12, 32, 65, 72, 42, 8, 0, 4, 12, 34, 70, 106, 77, 28, 1, 1, 2, 12, 40, 90, 150

Offset: 2

Emeric Deutsch, Aug 06 2006

Row n has floor(n/2) terms.
Sum of terms in row n = A023360(n).
T(n,1) = A010051(n) (characteristic function of primes); T(n,2) = A073610(n); T(n,3) = A098238(n).
Sum_{k=1..floor(n/2)} k*T(n,k) = A121304(n).

			T(9,3) = 4 because we have [2,2,5], [2,5,2], [5,2,2] and [3,3,3].
Triangle starts:
  1;
  1;
  0, 1;
  1, 2;
  0, 1, 1;
  1, 2, 3;
  0, 2, 3, 1;
  0, 2, 4, 4;
  ...

Alois P. Heinz, Rows n = 2..200, flattened

Cf. A010051, A023360, A073610, A098238, A121304, A224344.

T(n^2,n) gives A341459.

G:=1/(1-t*sum(z^ithprime(i),i=1..30))-1: Gser:=simplify(series(G,z=0,25)): for n from 2 to 21 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 2 to 21 do seq(coeff(P[n],t,j),j=1..floor(n/2)) od; # yields sequence in triangular form
# second Maple program:
with(numtheory):
b:= proc(n) option remember; local j; if n=0 then [1]
      else []; for j to pi(n) do zip((x, y)->x+y, %,
      [0, b(n-ithprime(j))[]], 0) od; % fi
    end:
T:= n-> subsop(1=NULL, b(n))[]:
seq(T(n), n=2..20);  # Alois P. Heinz, May 23 2013

nn=20;a[x_]:=Sum[x^Prime[n],{n,1,nn}];CoefficientList[Series[1/(1-y a[x]),{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Nov 08 2013 *)

G.f.: 1/(1 - t*Sum_{i>=1} z^prime(i)).

A282064 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^3.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 10, 15, 18, 22, 27, 33, 37, 45, 48, 52, 54, 60, 60, 69, 69, 79, 81, 87, 79, 93, 87, 97, 99, 114, 99, 120, 111, 130, 126, 150, 135, 168, 141, 160, 147, 177, 144, 189, 156, 183, 162, 201, 157, 213, 171, 214, 189, 231, 168, 237, 189, 244, 201, 261, 177, 270, 201, 261, 210, 282, 192, 297, 216, 283, 228

Offset: 0

Ilya Gutkovskiy, Feb 05 2017

nonn

Number of ways to write n as an ordered sum of three prime powers (1 included).

			a(6) = 10 because we have  [4, 1, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 2, 2], [2, 1, 3], [1, 4, 1], [1, 3, 2], [1, 2, 3] and [1, 1, 4].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power

Cf. A000961, A098238, A280243, A282062.

nmax = 70; CoefficientList[Series[(x + Sum[Floor[1/PrimeNu[k]] x^k, {k, 2, nmax}])^3, {x, 0, nmax}], x]

G.f.: (x + Sum_{p prime, k>=1} x^(p^k))^3.

A340960 Number of ways to write n as an ordered sum of 4 primes.

Original entry on oeis.org

1, 4, 6, 8, 13, 16, 22, 24, 22, 32, 34, 40, 47, 48, 56, 68, 70, 76, 90, 84, 111, 112, 126, 120, 144, 120, 176, 140, 184, 148, 226, 168, 264, 184, 262, 196, 313, 192, 352, 208, 366, 256, 418, 240, 473, 260, 496, 324, 536, 300, 616, 308, 634, 348, 670, 348, 772, 364, 786, 412

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..10000

Column k=4 of A121303.

Cf. A000040, A010051, A073610, A098238, A259194, A340961, A340962, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 4):
seq(a(n), n=8..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^4, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (Sum_{k>=1} x^prime(k))^4.

A340961 Number of ways to write n as an ordered sum of 5 primes.

Original entry on oeis.org

1, 5, 10, 15, 25, 36, 50, 65, 70, 90, 110, 125, 155, 170, 200, 241, 270, 300, 350, 375, 435, 500, 530, 600, 640, 696, 760, 850, 840, 985, 990, 1170, 1160, 1370, 1250, 1570, 1445, 1760, 1600, 2000, 1710, 2340, 1950, 2555, 2165, 2876, 2320, 3340, 2560, 3595, 2880, 3985, 3050

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 10..10000

Column k=5 of A121303.

Cf. A000040, A010051, A073610, A098238, A259195, A340960, A340962, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 5):
seq(a(n), n=10..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (Sum_{k>=1} x^prime(k))^5.

A340962 Number of ways to write n as an ordered sum of 6 primes.

Original entry on oeis.org

1, 6, 15, 26, 45, 72, 106, 150, 186, 236, 306, 366, 455, 540, 636, 782, 912, 1056, 1236, 1410, 1617, 1896, 2106, 2400, 2696, 2976, 3348, 3716, 4026, 4446, 4917, 5340, 5982, 6380, 7017, 7476, 8377, 8640, 9765, 9936, 11202, 11496, 13132, 12930, 15117, 14672, 17178, 16800, 19696

Offset: 12

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 12..10000

Column k=6 of A121303.

Cf. A000040, A010051, A073610, A098238, A259196, A340960, A340961, A340963, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 6):
seq(a(n), n=12..60);  # Alois P. Heinz, Jan 31 2021

nmax = 60; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^6, {x, 0, nmax}], x] // Drop[#, 12] &

G.f.: (Sum_{k>=1} x^prime(k))^6.

A340963 Number of ways to write n as an ordered sum of 7 primes.

Original entry on oeis.org

1, 7, 21, 42, 77, 133, 210, 316, 434, 574, 770, 980, 1239, 1547, 1876, 2331, 2828, 3367, 4032, 4746, 5565, 6574, 7602, 8757, 10136, 11480, 13132, 14882, 16646, 18662, 20951, 23268, 26082, 28861, 31787, 35218, 38745, 42532, 46403, 50883, 54810, 60613, 65016, 71302, 76069

Offset: 14

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 14..10000

Column k=7 of A121303.

Cf. A000040, A010051, A073610, A098238, A259197, A340960, A340961, A340962, A340964, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 7):
seq(a(n), n=14..58);  # Alois P. Heinz, Jan 31 2021

nmax = 58; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 14] &

G.f.: (Sum_{k>=1} x^prime(k))^7.

A340964 Number of ways to write n as an ordered sum of 8 primes.

Original entry on oeis.org

1, 8, 28, 64, 126, 232, 392, 624, 925, 1296, 1800, 2416, 3158, 4088, 5152, 6504, 8142, 9976, 12216, 14784, 17738, 21296, 25272, 29736, 35023, 40768, 47328, 54832, 62728, 71744, 81796, 92736, 105078, 118664, 132924, 149424, 167002, 186144, 206852, 229272, 253023

Offset: 16

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 16..10000

Column k=8 of A121303.

Cf. A000040, A010051, A073610, A098238, A259198, A340960, A340961, A340962, A340963, A340965, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 8):
seq(a(n), n=16..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^8, {x, 0, nmax}], x] // Drop[#, 16] &

G.f.: (Sum_{k>=1} x^prime(k))^8.

A340965 Number of ways to write n as an ordered sum of 9 primes.

Original entry on oeis.org

1, 9, 36, 93, 198, 387, 696, 1170, 1845, 2740, 3960, 5562, 7566, 10125, 13248, 17133, 22014, 27774, 34776, 43173, 53010, 64869, 78696, 94617, 113415, 134946, 159552, 188164, 219960, 256041, 297180, 342846, 394614, 452595, 516276, 587997, 667938, 755109, 852444

Offset: 18

Ilya Gutkovskiy, Jan 31 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 18..10000

Column k=9 of A121303.

Cf. A000040, A010051, A073610, A098238, A259200, A340960, A340961, A340962, A340963, A340964, A340966.

b:= proc(n, k) option remember; local r, p; r, p:= 0, 2;
      if n=0 then `if`(k=0, 1, 0) elif k<1 then 0 else
      while p<=n do r:= r+b(n-p, k-1); p:= nextprime(p) od; r fi
    end:
a:= n-> b(n, 9):
seq(a(n), n=18..56);  # Alois P. Heinz, Jan 31 2021

nmax = 56; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}]^9, {x, 0, nmax}], x] // Drop[#, 18] &

G.f.: (Sum_{k>=1} x^prime(k))^9.

cp's OEIS Frontend

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

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A073610 Number of primes of the form n-p where p is a prime.

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A121303 Triangle read by rows: T(n,k) is the number of compositions of n into k primes (i.e., ordered sequences of k primes having sum n; n>=2, k>=1).

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A282064 Expansion of (x + Sum_{p prime, k>=1} x^(p^k))^3.

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A340960 Number of ways to write n as an ordered sum of 4 primes.

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A340961 Number of ways to write n as an ordered sum of 5 primes.

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A340962 Number of ways to write n as an ordered sum of 6 primes.

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