A014612 -id:A014612 - cp's OEIS frontend

A098238 Number of ordered ways of writing n as sum of three primes.

0, 0, 0, 0, 0, 0, 1, 3, 3, 4, 6, 6, 9, 6, 6, 10, 9, 12, 12, 12, 12, 19, 12, 21, 15, 21, 18, 30, 15, 30, 12, 30, 18, 37, 12, 39, 21, 42, 24, 46, 9, 51, 18, 48, 24, 54, 18, 66, 21, 60, 30, 67, 24, 81, 18, 75, 30, 79, 18, 87, 21, 87, 36, 93, 15, 105, 30, 105, 36, 97, 12, 120, 30, 114, 36

Offset: 0

Ralf Stephan, Aug 31 2004

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 213. [?Broken link]
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 213.

Cf. A073610, A014612.

Cf. A068307.

t1:=add(q^ithprime(n),n=1..1000): series(t1^3,q,1001): seriestolist(%); # N. J. A. Sloane, Sep 29 2006

nn = 74; a = Sum[x^p, {p, Prime[Range[nn]]}]; CoefficientList[Series[a^3, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 25 2015 *)

G.f.: (Sum_{k>0} x^prime(k))^3. - Vladeta Jovovic, Mar 12 2005

Third convolution of "a(n)=1 if n prime, 0 otherwise" (A010051) with itself. - Graeme McRae, Jul 18 2006

A101271 Number of partitions of n into 3 distinct and relatively prime parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 8, 9, 12, 12, 16, 15, 21, 20, 26, 25, 33, 28, 40, 36, 45, 42, 56, 44, 65, 56, 70, 64, 84, 66, 96, 81, 100, 88, 120, 90, 133, 110, 132, 121, 161, 120, 175, 140, 176, 156, 208, 153, 220, 180, 222, 196, 261, 184, 280, 225, 270, 240, 312, 230, 341, 272

Offset: 6

Vladeta Jovovic, Dec 19 2004

The Heinz numbers of these partitions are the intersection of A289509 (relatively prime), A005117 (strict), and A014612 (triple). - Gus Wiseman, Oct 15 2020

			For n=10 we have 4 such partitions: 1+2+7, 1+3+6, 1+4+5 and 2+3+5.
From _Gus Wiseman_, Oct 13 2020: (Start)
The a(6) = 1 through a(18) = 15 triples (A..F = 10..15):
  321  421  431  432  532  542  543  643  653  654  754  764  765
            521  531  541  632  651  652  743  753  763  854  873
                 621  631  641  732  742  752  762  853  863  954
                      721  731  741  751  761  843  871  872  972
                           821  831  832  851  852  943  953  981
                                921  841  932  861  952  962  A53
                                     931  941  942  961  971  A71
                                     A21  A31  951  A51  A43  B43
                                          B21  A32  B32  A52  B52
                                               A41  B41  A61  B61
                                               B31  C31  B42  C51
                                               C21  D21  B51  D32
                                                         C32  D41
                                                         C41  E31
                                                         D31  F21
                                                         E21
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000

Cf. A023024-A023030, A000742-A000743, A023031-A023035.
A000741 is the ordered non-strict version.
A001399(n-6) does not require relative primality.
A023022 counts pairs instead of triples.
A023023 is the not necessarily strict version.
A078374 counts these partitions of any length, with Heinz numbers A302796.
A101271*6 is the ordered version.
A220377 is the pairwise coprime instead of relatively prime version.
A284825 counts the case that is pairwise non-coprime also.
A337605 is the pairwise non-coprime instead of relatively prime version.
A008289 counts strict partitions by sum and length.
A007304 gives the Heinz numbers of 3-part strict partitions.
A307719 counts 3-part pairwise coprime partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A000837, A007360, A014612, A055684, A289509, A332004, A337452, A337563.

m:=3: with(numtheory): g:=sum(mobius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k),i=1..m),k=1..20): gser:=series(g,x=0,80): seq(coeff(gser,x^n),n=6..77); # Emeric Deutsch, May 31 2005

Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&GCD@@#==1&]],{n,6,50}] (* Gus Wiseman, Oct 13 2020 *)

G.f. for the number of partitions of n into m distinct and relatively prime parts is Sum(moebius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k), i=1..m), k=1..infinity).

More terms from Emeric Deutsch, May 31 2005

A337605 Number of unordered triples of distinct positive integers summing to n, any two of which have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 4, 1, 5, 0, 9, 0, 8, 3, 10, 0, 17, 1, 14, 5, 16, 1, 25, 1, 21, 8, 26, 2, 37, 1, 30, 15, 33, 2, 49, 2, 44, 16, 44, 2, 64, 6, 54, 21, 56, 3, 87, 5, 65, 30, 70, 9, 101, 5, 80, 34, 98, 6, 121, 6, 96, 52

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(n) triples for n = 12, 16, 18, 22, 27, 55:
  (6,4,2)  (8,6,2)   (8,6,4)   (10,8,4)  (12,9,6)  (28,21,6)
           (10,4,2)  (9,6,3)   (12,6,4)  (15,9,3)  (30,20,5)
                     (10,6,2)  (12,8,2)  (18,6,3)  (35,15,5)
                     (12,4,2)  (14,6,2)            (40,10,5)
                               (16,4,2)            (25,20,10)
                                                   (30,15,10)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014612 intersected with A318719 ranks these partitions.
A220377 is the coprime instead of non-coprime version.
A318717 counts these partitions of any length, ranked by A318719.
A337599 is the non-strict version.
A337604 is the ordered non-strict version.
A337605*6 is the ordered version.
A023023 counts relatively prime 3-part partitions
A051424 counts pairwise coprime or singleton partitions.
A200976 and A328673 count pairwise non-coprime partitions.
A307719 counts pairwise coprime 3-part partitions.
A327516 counts pairwise coprime partitions, with strict case A305713.
Cf. A000217, A001399, A014612, A082024, A178472, A220377, A284825, A337461, A337561, A337667.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,100}]

A065119 Numbers k such that the k-th cyclotomic polynomial is a trinomial.

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, 192, 216, 243, 288, 324, 384, 432, 486, 576, 648, 729, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2187, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6561, 6912, 7776, 8748, 9216

Offset: 1

Len Smiley, Nov 12 2001

nonn

Appears to be numbers of form 2^a * 3^b, a >= 0, b > 0. - Lekraj Beedassy, Sep 10 2004
This is true: see link "Cyclotomic trinomials". - Robert Israel, Jul 14 2015
3-smooth numbers (A003586) which are not powers of 2 (A000079). - Amiram Eldar, Nov 10 2020
These are the conjugates of semiprimes, where conjugation is A122111; or Heinz numbers of conjugates of length-2 partitions. - Gus Wiseman, Nov 09 2023
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 13 2024

			The 54th cyclotomic polynomial is x^18 - x^9 + 1 which is trinomial, so 54 is in the sequence.
From _Gus Wiseman_, Nov 09 2023: (Start)
The terms and conjugate semiprimes, showing their respective Heinz partitions, begin:
    3: (2)              4: (1,1)
    6: (2,1)            6: (2,1)
    9: (2,2)            9: (2,2)
   12: (2,1,1)         10: (3,1)
   18: (2,2,1)         15: (3,2)
   24: (2,1,1,1)       14: (4,1)
   27: (2,2,2)         25: (3,3)
   36: (2,2,1,1)       21: (4,2)
   48: (2,1,1,1,1)     22: (5,1)
   54: (2,2,2,1)       35: (4,3)
   72: (2,2,1,1,1)     33: (5,2)
   81: (2,2,2,2)       49: (4,4)
   96: (2,1,1,1,1,1)   26: (6,1)
(End)

Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 733, pp. 74 and 310, Ellipses Paris, 2004.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Cyclotomic trinomials

Differs at the 18th term from A063996.
Cf. A003586, A033845, A206787.
For primes (A008578) we have conjugates A000079.
For triprimes (A014612) we have conjugates A080193.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A000040, A000720, A001248, A046682, A056239, A086971, A122111, A220264.

with(numtheory): a := []; for m from 1 to 3000 do if nops([coeffs(cyclotomic(m,x))])=3 then a := [op(a),m] fi od; print(a);

max = 5000; Sort[Flatten[Table[2^a 3^b, {a, 0, Floor[Log[2, max]]}, {b, Floor[Log[3, max/2^a]]}]]] (* Alonso del Arte, May 19 2016 *)

isok(n)=my(vp = Vec(polcyclo(n))); sum(k=1, #vp, vp[k] != 0) == 3; \\ Michel Marcus, Jul 11 2015

list(lim)=my(v=List(),N); for(n=1,logint(lim\1,3), N=3^n; while(N<=lim, listput(v,N); N<<=1)); Set(v) \\ Charles R Greathouse IV, Aug 07 2015

A206787(a(n)) = 4. - Reinhard Zumkeller, Feb 12 2012
a(n) = A033845(n)/2 = 3 * A003586(n). - Robert Israel, Jul 14 2015
Sum_{n>=1} 1/a(n) = 1. - Amiram Eldar, Nov 10 2020

Offset set to 1 and more terms from Michel Marcus, Jul 11 2015

A337563 Number of pairwise coprime unordered triples of positive integers > 1 summing to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 7, 1, 7, 3, 9, 2, 15, 3, 13, 5, 17, 4, 29, 5, 20, 8, 28, 8, 42, 8, 31, 14, 42, 10, 59, 12, 45, 21, 52, 14, 77, 17, 68, 26, 69, 19, 101, 26, 84, 34, 86, 25, 138, 28, 95, 43, 111, 36, 161, 35, 118, 52, 151

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

Such partitions are necessarily strict.

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A014612 (triples), and A302696 (coprime).

			The a(10) = 1 through a(24) = 15 triples (empty columns indicated by dots, A..J = 10..19):
  532  .  543  .  743  753  754  .  765  B53  875  975  985  B75  987
          732     752       853     873       974  B73  B65  D73  B76
                            952     954       A73  D53  B74       B85
                            B32     972       B54       B83       B94
                                    B43       B72       B92       BA3
                                    B52       D43       D54       C75
                                    D32       D52       D72       D65
                                                        E53       D74
                                                        H32       D83
                                                                  D92
                                                                  F72
                                                                  G53
                                                                  H43
                                                                  H52
                                                                  J32

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A055684 is the version for pairs.
A220377 allows 1's, with non-strict version A307719.
A337485 counts these partitions of any length.
A337563*6 is the ordered version.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 counts pairwise coprime partitions with no 1's.
A078374 counts relatively prime strict partitions.
A200976 and A328673 count pairwise non-coprime partitions.
A302696 ranks pairwise coprime partitions.
A302698 counts relatively prime partitions with no 1's.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A337452 counts relatively prime strict partitions with no 1's.
Cf. A007304, A082024, A101268, A284825, A332004, A337451, A337461, A337462, A337561, A337599, A337601, A337605.

Table[Length[Select[IntegerPartitions[n,{3}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]

A337604 Number of ordered triples of positive integers summing to n, any two of which have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 6, 0, 13, 0, 15, 7, 21, 0, 37, 0, 39, 16, 45, 0, 73, 6, 66, 28, 81, 0, 130, 6, 105, 46, 120, 21, 181, 6, 153, 67, 189, 12, 262, 6, 213, 118, 231, 12, 337, 21, 306, 121, 303, 12, 433, 57, 369, 154, 378, 18, 583, 30, 435, 217, 465

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

The first relatively prime triple (15,10,6) is counted under a(31).

			The a(6) = 1 through a(15) = 7 triples (empty columns indicated by dots, A = 10):
  222  .  224  333  226  .  228  .  22A  339
          242       244     246     248  366
          422       262     264     266  393
                    424     282     284  555
                    442     336     2A2  636
                    622     363     428  663
                            426     446  933
                            444     464
                            462     482
                            624     626
                            633     644
                            642     662
                            822     824
                                    842
                                    A22

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014311 intersected with A337666 ranks these compositions.
A337667 counts these compositions of any length.
A335402 lists the positions of zeros.
A337461 is the coprime instead of non-coprime version.
A337599 is the unordered version, with strict case A337605.
A337605*6 is the strict version.
A000741 counts relatively prime 3-part compositions.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 count pairwise non-relatively prime partitions.
A307719 counts pairwise coprime 3-part partitions.
A318717 counts pairwise non-coprime strict partitions.
A333227 ranks pairwise coprime compositions.
Cf. A000217, A001399, A014612, A051424, A082024, A178472, A220377, A284825, A305713, A327516, A333228, A337561.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]

A109251 Number of numbers up to 10^n which are products of three primes.

Original entry on oeis.org

0, 1, 22, 247, 2569, 25556, 250853, 2444359, 23727305, 229924367, 2227121996, 21578747909, 209214982913, 2030133769624, 19717814526785, 191693417109381, 1865380637252270, 18168907486812690, 177123437184971927, 1728190923820610000

Offset: 0

Martin Raab, Aug 19 2005

nonn

			There are 22 numbers with three prime factors up to 10^2: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99.

Cf. A014612 = numbers with three prime factors, A036352 = number of numbers up to 10^n which are products of two primes, A072114.

ThreeAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@ Sqrt[n/Prime@i]}]; Table[ ThreeAlmostPrimePi[10^n], {n, 0, 14}] (* Robert G. Wilson v *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A109251(n):
    r = 10**n
    return int(sum(primepi(r//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(r,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(r//k)+1),a))) # Chai Wah Wu, Sep 18 2024

a(n) = A072114(10^n). - R. J. Mathar, May 25 2008

a(10)-a(14) from Robert G. Wilson v, Feb 06 2006
a(15)-a(17) from Hiroaki Yamanouchi, Aug 30 2014
a(18)-a(19) from Henri Lifchitz, Dec 01 2014

A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 11, 13, 17, 20, 24, 28, 33, 37, 43, 48, 54, 60, 67, 73, 81, 88, 96, 104, 113, 121, 131, 140, 150, 160, 171, 181, 193, 204, 216, 228, 241, 253, 267, 280, 294, 308, 323, 337, 353, 368, 384, 400, 417, 433, 451, 468, 486, 504, 523, 541, 561, 580, 600

Offset: 0

Jack W Grahl, Feb 02 2009, Feb 11 2009

For n = 1, 2 these are just the triangular numbers. a(n) is always at least 1/3 of the corresponding triangular number, since each partition of this type gives up to three ordered partitions with the same cyclical order.
An alternative definition, which avoids using parts of size 0: a(n) is the third diagonal of A184957. - N. J. A. Sloane, Feb 27 2011
Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending m-nomial triangles (see A004737):
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4 5 4 3 2 1
1 2 3 4 5 6 5 4 3 2 1
- Bob Selcoe, Feb 07 2014
Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:
1  1  2  3  3  4  5  5  6 ...
      1  1  2  3  3  4  5 ...
            1  1  2  3  3 ...
                  1  1  2 ...
                        1 ...
------------------------------
1  1  3  4  6  8 11 13 17 ... - L. Edson Jeffery, Jul 30 2014
a(n) is the dimension of three-dimensional (2n + 2)-homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal. - Giedrius Alkauskas, Sep 30 2017
Also the number of compositions of n + 3 into three parts, the first at least as great as each of the other two. Also the number of compositions of n + 4 into three parts, the first strictly greater than each of the other two. - Gus Wiseman, Oct 09 2020

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
From _Gus Wiseman_, Oct 05 2020: (Start)
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
  (000)  (100)  (101)  (111)  (202)  (212)  (222)  (313)
                (110)  (201)  (211)  (221)  (303)  (322)
                (200)  (210)  (220)  (302)  (312)  (331)
                       (300)  (301)  (311)  (321)  (403)
                              (310)  (320)  (330)  (412)
                              (400)  (401)  (402)  (421)
                                     (410)  (411)  (430)
                                     (500)  (420)  (502)
                                            (501)  (511)
                                            (510)  (520)
                                            (600)  (601)
                                                   (610)
                                                   (700)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG] (2017), Section 4.3.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

For compositions into 4 summands see A156039; also see A156041 and A156042.
Cf. A184957, A071619 (bisection).
A001399(n-2)*2 is the strict case.
A001840(n-2) is the version with opposite relations.
A001840(n-1) is the version with strict opposite relations.
A069905 is the case with strict relations.
A014311 ranks 3-part compositions, with strict case A337453.
A014612 ranks 3-part partitions, with strict case A007304.
Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.

a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009

nn = 58; CoefficientList[Series[x^3/(1 - x^2)^2/(1 - x^3) + 1/(1 - x^2)^2/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3,{3}],#[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020*)

{a(n) = n*(n+4)\6 + 1}; /* Michael Somos, Mar 26 2017 */

G.f.: (x^2+1) / (1-x-x^2+x^4+x^5-x^6). - Alois P. Heinz, Jun 14 2009
Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^3)). - N. J. A. Sloane, Apr 29 2011
a(n) = A007590(n+2) - A000212(n+2). - Richard R. Forberg, Dec 08 2013
a(2*n) = A071619(n+1). - L. Edson Jeffery, Jul 29 2014
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 6, a(5) = 8. - Harvey P. Dale, May 28 2015
a(n) = (n^2 + 4*n + 3)/6 + IF(MOD(n, 2) = 0, 1/2) + IF(MOD(n, 3) = 1, -1/3). - Heinrich Ludwig, Mar 21 2017
a(n) = 1 + floor((n^2 + 4*n)/6). - Giovanni Resta, Mar 21 2017
Euler transform of length 4 sequence [1, 2, 1, -1]. - Michael Somos, Mar 26 2017
a(n) = a(-4 - n) for all n in Z. - Michael Somos, Mar 26 2017
0 = a(n)*(-1 + a(n) - 2*a(n+1) - 2*a(n+2) + 2*a(n+3)) + a(n+1)*(+1 + a(n+1) + 2*a(n+2) - 2*a(n+3)) + a(n+2)*(+1 + a(n+2) - 2*a(n+3)) + a(n+3)*(-1 + a(n+3)) for all n in Z. - Michael Somos, Mar 26 2017
a(n) = round((n+1)*(n+3)/6). - Bill McEachen, Feb 16 2021
Sum_{n>=0} 1/a(n) = 3/2 + Pi^2/36 + (tan(c1)-1)*c1 + 3*c2*sinh(c2)/(1+2*cosh(c2)), where c1 = Pi/(2*sqrt(3)) and c2 = Pi*sqrt(2)/3. - Amiram Eldar, Dec 10 2022
E.g.f.: ((16 + 15*x + 3*x^2)*cosh(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (7 + 15*x + 3*x^2)*sinh(x))/18. - Stefano Spezia, Apr 05 2023

More terms from Alois P. Heinz, Jun 14 2009

A101637 a(n) = 1 if n is a 4-almost prime, that is a product of exactly four (not necessarily distinct) primes, 0 otherwise.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Jonathan Vos Post, Dec 10 2004

Characteristic function of A014613.

See A101638 for the inverse Moebius transform of this sequence.

			a(100) = 1 because 100 = 2 * 2 * 5 * 5 is the product of exactly 4 primes and thus is a 4-almost prime.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Almost Prime.
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A101638, A014613, A000040, A001358, A014612, A014614.

Table[If[PrimeOmega[n]==4,1,0],{n,100}] (* Harvey P. Dale, Sep 13 2024 *)

a(n)=bigomega(n)==4 \\ Charles R Greathouse IV, Jan 31 2017

Name edited by Antti Karttunen, Oct 08 2017

A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

Original entry on oeis.org

1, 3, 9, 20, 54, 112, 240, 648, 1344, 2816, 5760, 12800, 26624, 62208, 129024, 270336, 552960, 1114112, 2293760, 4915200, 9961472, 20447232, 47775744, 96468992, 198180864, 411041792, 830472192, 1698693120, 3422552064, 7046430720

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

nonn

A k-almost prime is a positive integer that has exactly k prime factors counted with multiplicity.

			a(0) = 1 since one is the multiplicative identity,
a(1) = 2nd 1-almost prime is the second prime number = A000040(2) = 3,
a(2) = 3rd 2-almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.
a(3) = 4th 3-almost prime = A014612(4) = 20 = {2*2*5}.
a(4) = 5th 4-almost prime = A014613(5) = 54 = {2*3*3*3},
a(5) = 6th 5-almost prime = A014614(6) = 112 = {2*2*2*2*7}, ....

Chai Wah Wu, Table of n, a(n) for n = 0..1000 (terms 0..228 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078842, A078843, A078844, A078445, A078846, A101695.

f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[ t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Table[ t[[n, n + 1]], {n, 30}] (* Robert G. Wilson v, Feb 11 2006 *)
AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[ Array[a,i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n-1, n]], {n, 30}]; lst (* Robert G. Wilson v, Nov 13 2007 *)

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A078841(n):
    if n <= 1: return (n<<1)+1
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Conjecture: Lim as n->inf. of a(n+1)/a(n) = 2. - Robert G. Wilson v, Nov 13 2007

a(14)-a(29) from Robert G. Wilson v, Feb 11 2006

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