A284825 -id:A284825 - cp's OEIS frontend

A007304 Sphenic numbers: products of 3 distinct primes.

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

A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 9, 3, 15, 9, 21, 9, 39, 9, 45, 21, 45, 21, 87, 21, 93, 39, 87, 39, 153, 39, 135, 63, 153, 57, 255, 51, 207, 93, 225, 93, 321, 81, 291, 135, 321, 105, 471, 105, 393, 183, 381, 147, 597, 147, 531, 213, 507, 183, 759, 207, 621, 273, 621, 231

Offset: 0

Gus Wiseman, Sep 02 2020

nonn

			The a(3) = 1 through a(9) = 9 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
           (1,2,1)  (1,3,1)  (1,2,3)  (1,5,1)  (1,2,5)  (1,3,5)
           (2,1,1)  (3,1,1)  (1,3,2)  (5,1,1)  (1,3,4)  (1,5,3)
                             (1,4,1)           (1,4,3)  (1,7,1)
                             (2,1,3)           (1,5,2)  (3,1,5)
                             (2,3,1)           (1,6,1)  (3,5,1)
                             (3,1,2)           (2,1,5)  (5,1,3)
                             (3,2,1)           (2,5,1)  (5,3,1)
                             (4,1,1)           (3,1,4)  (7,1,1)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

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

A000212 counts the unimodal instead of coprime version.
A220377*6 is the strict case.
A307719 is the unordered version.
A337462 counts these compositions of any length.
A337563 counts the case of partitions with no 1's.
A337603 only requires the *distinct* parts to be pairwise coprime.
A337604 is the intersecting instead of coprime version.
A014612 ranks 3-part partitions.
A302696 ranks pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000217, A001399, A001840, A014311, A101268, A284825, A337562, A326675, A333227, A337601, A337602.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,30}]

A307719 Number of partitions of n into 3 mutually coprime parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 2, 7, 2, 8, 4, 8, 4, 15, 4, 16, 7, 15, 7, 26, 7, 23, 11, 26, 10, 43, 9, 35, 16, 38, 16, 54, 14, 49, 23, 54, 18, 79, 18, 66, 31, 64, 25, 100, 25, 89, 36, 85, 31, 127, 35, 104, 46, 104, 39, 167, 36, 125, 58, 129, 52, 185, 45

Offset: 0

Wesley Ivan Hurt, Apr 24 2019

The Heinz numbers of these partitions are the intersection of A014612 (triples) and A302696 (pairwise coprime). - Gus Wiseman, Oct 16 2020

			There are 2 partitions of 9 into 3 mutually coprime parts: 7+1+1 = 5+3+1, so a(9) = 2.
There are 4 partitions of 10 into 3 mutually coprime parts: 8+1+1 = 7+2+1 = 5+4+1 = 5+3+2, so a(10) = 4.
There are 2 partitions of 11 into 3 mutually coprime parts: 9+1+1 = 7+3+1, so a(11) = 2.
There are 7 partitions of 12 into 3 mutually coprime parts: 10+1+1 = 9+2+1 = 8+3+1 = 7+4+1 = 6+5+1 = 7+3+2 = 5+4+3, so a(12) = 7.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)
Index entries for sequences related to partitions

A023022 is the version for pairs.
A220377 is the strict case, with ordered version A220377*6.
A327516 counts these partitions of any length, with strict version A305713 and Heinz numbers A302696.
A337461 is the ordered version.
A337563 is the case with no 1's.
A337599 is the pairwise non-coprime instead of pairwise coprime version.
A337601 only requires the distinct parts to be pairwise coprime.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A007304, A014612, A078374, A082024, A284825, A304709, A337462, A337605.

N:= 200: # to get a(0)..a(N)
A:= Array(0..N):
for a from 1 to N/3 do
  for b from a to (N-a)/2 do
    if igcd(a,b) > 1 then next fi;
    ab:= a*b;
    for c from b to N-a-b do
       if igcd(ab,c)=1 then A[a+b+c]:= A[a+b+c]+1 fi
od od od:
convert(A,list); # Robert Israel, May 09 2019

Table[Sum[Sum[Floor[1/(GCD[i, j] GCD[j, n - i - j] GCD[i, n - i - j])], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,100}] (* Gus Wiseman, Oct 15 2020 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [gcd(i,j) * gcd(j,n-i-j) * gcd(i,n-i-j) = 1], where [] is the Iverson bracket.

a(n > 2) = A220377(n) + 1. - Gus Wiseman, Oct 15 2020

A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290

Offset: 0

Clark Kimberling, Apr 15 2012

For a guide to related sequences, see A211422.
Also the number of partitions of n+1 into three parts, where each part > 1. - Peter Woodward, May 25 2015
a(n) is also equal to the number of partitions of n+4 into three distinct parts, where each part > 1. - Giovanni Resta, May 26 2015
Number of different distributions of n+1 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Dec 31 2015
After the first three terms, partial sums of A008615. - Robert Israel, Dec 31 2015
For n >= 2, also the number of partitions of n - 2 into 3 parts. The Heinz numbers of these partitions are given by A014612. - Gus Wiseman, Oct 11 2020

			a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015
a(11) = 5 Number of different distributions of 11 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
a(1) = a(2) = a(3) = a(4) = a(5) = 0, since with fewer than 6 identical balls there is no such distribution with 3 boxes that holds for 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...
From _Gus Wiseman_, Oct 11 2020: (Start)
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three parts > 1 [Woodward] are the following (A = 10, B = 11, C = 12). The ordered version is A000217(n - 4) and the Heinz numbers are A046316.
  222  322  332  333  433  443  444  544  554  555  655
            422  432  442  533  543  553  644  654  664
                 522  532  542  552  643  653  663  754
                      622  632  633  652  662  744  763
                           722  642  733  743  753  772
                                732  742  752  762  844
                                822  832  833  843  853
                                     922  842  852  862
                                          932  933  943
                                          A22  942  952
                                               A32  A33
                                               B22  A42
                                                    B32
                                                    C22
The a(5) = 1 through a(15) = 14 partitions of n + 4 into three distinct parts > 1 [Resta] are the following (A = 10, B = 11, C = 12, D = 13, E = 14). The ordered version is A211540*6 and the Heinz numbers are A046389.
  432  532  542  543  643  653  654  754  764  765  865
            632  642  652  743  753  763  854  864  874
                 732  742  752  762  853  863  873  964
                      832  842  843  862  872  954  973
                           932  852  943  953  963  982
                                942  952  962  972  A54
                                A32  A42  A43  A53  A63
                                     B32  A52  A62  A72
                                          B42  B43  B53
                                          C32  B52  B62
                                               C42  C43
                                               D32  C52
                                                    D42
                                                    E32
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three distinct parts [Uslu and Becenen] are the following (A = 10, B = 11, C = 12, D = 13). The ordered version is A211540(n)*6 and the Heinz numbers are A007304.
  321  421  431  432  532  542  543  643  653  654  754
            521  531  541  632  642  652  743  753  763
                 621  631  641  651  742  752  762  853
                      721  731  732  751  761  843  862
                           821  741  832  842  852  871
                                831  841  851  861  943
                                921  931  932  942  952
                                     A21  941  951  961
                                          A31  A32  A42
                                          B21  A41  A51
                                               B31  B32
                                               C21  B41
                                                    C31
                                                    D21
(End)

Robert Israel, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Ece Uslu and Esin Becenen, Identical Object Distributions.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A069905, A211422.
All of the following pertain to 3-part strict partitions.
- A000009 counts these partitions of any length, with non-strict version A000041.
- A007304 gives the Heinz numbers, with non-strict version A014612.
- A101271 counts the relatively prime case, with non-strict version A023023.
- A220377 counts the pairwise coprime case, with non-strict version A307719.
- A337605 counts the pairwise non-coprime case, with non-strict version A337599.
Cf. A000217, A001840, A156040, A284825, A337453, A337483, A337484, A337563.

I:=[0,0,0,0,0,1]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

f:= gfun:-rectoproc({a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6),seq(a(i)=0,i=0..4),a(5)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Dec 31 2015

t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211540 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = round( (n-2)^2 / 12 )}; / * Michael Somos, Feb 02 2015 */

concat(vector(5), Vec(x^5/(1-x-x^2+x^4+x^5-x^6) + O(x^100))) \\ Altug Alkan, Jan 10 2016

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).
a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - Alois P. Heinz, Nov 03 2012
a(n) = 3*k^2-6*k+3 (for n = 6*k-3), 3*k^2-5*k+2 (for n = 6*k-2), 3*k^2-4*k+1 (for n = 6*k-1), 3*k^2-3*k+1 (for n = 6*k), 3*k^2-2*k (for n = 6*k+1), 3*k^2-k (for n = 6*k+2). - Ece Uslu, Esin Becenen, Dec 31 2015
a(n) = A004526(n-2) + a(n-2) for n > 2. - Ece Uslu, Esin Becenen, Dec 31 2015
G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - Robert Israel, Dec 31 2015
a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - Wesley Ivan Hurt, Apr 27 2019
From Gus Wiseman, Oct 11 2020: (Start)
a(n+2) = A069905(n) = A001399(n-3) counts 3-part partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part strict partitions.
a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part partitions with no 1's.
a(n-4) = A069905(n-6) = A001399(n-9) counts 3-part strict partitions with no 1's.
A000217(n-2) counts 3-part compositions.
a(n-1)*6 = A069905(n-3)*6 = A001399(n-6)*6 counts 3-part strict compositions.
A000217(n-5) counts 3-part compositions with no 1's.
a(n-4)*6 = A069905(n-6)*6 = A001399(n-9)*6 counts 3-part strict compositions with no 1's.
(End)

A023023 Number of partitions of n into 3 unordered relatively prime parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 10, 8, 14, 12, 16, 16, 24, 18, 30, 24, 32, 30, 44, 32, 50, 42, 54, 48, 70, 48, 80, 64, 80, 72, 96, 72, 114, 90, 112, 96, 140, 96, 154, 120, 144, 132, 184, 128, 196, 150, 192, 168, 234, 162, 240, 192, 240, 210, 290, 192, 310, 240, 288, 256, 336, 240, 374

Offset: 3

David W. Wilson

nonn

			From _Gus Wiseman_, Oct 08 2020: (Start)
The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11):
  111   211   221   321   322   332   432   433   443   543   544
              311   411   331   431   441   532   533   552   553
                          421   521   522   541   542   651   643
                          511   611   531   631   551   732   652
                                      621   721   632   741   661
                                      711   811   641   831   733
                                                  722   921   742
                                                  731   A11   751
                                                  821         832
                                                  911         841
                                                              922
                                                              931
                                                              A21
                                                              B11
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 3..10000
Mohamed El Bachraoui, Relatively Prime Partitions with Two and Three Parts, Fibonacci Quart. 46/47 (2008/2009), no. 4, 341-345.

Cf. A023024-A023030, A000742-A000743, A023032-A023035.
A000741 is the ordered version.
A000837 counts these partitions of any length.
A001399(n-3) does not require relative primality.
A023022 is the 2-part version.
A101271 is the strict case.
A284825 counts the case that is also pairwise non-coprime.
A289509 intersected with A014612 gives the Heinz numbers.
A307719 is the pairwise coprime instead of relatively prime version.
A337599 is the pairwise non-coprime instead of relative prime version.
A008284 counts partitions by sum and length.
A078374 counts relatively prime strict partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A007434, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605.

Table[Length[Select[IntegerPartitions[n,{3}],GCD@@#==1&]],{n,3,50}] (* Gus Wiseman, Oct 08 2020 *)

G.f. for the number of partitions of n into m unordered relatively prime parts is Sum(moebius(k)*x^(m*k)/Product(1-x^(i*k), i=1..m), k=1..infinity). - Vladeta Jovovic, Dec 21 2004
a(n) = (n^2/12)*Product_{prime p|n} (1 - 1/p^2) = A007434(n)/12 for n > 3 (proved by Mohamed El Bachraoui). [Jonathan Sondow, May 27 2009]
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,k,n-i-k)). - Wesley Ivan Hurt, Jan 02 2021

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}]

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}]

A000741 Number of compositions of n into 3 ordered relatively prime parts.

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 15, 18, 27, 30, 45, 42, 66, 63, 84, 84, 120, 99, 153, 132, 174, 165, 231, 180, 270, 234, 297, 270, 378, 276, 435, 360, 450, 408, 540, 414, 630, 513, 636, 552, 780, 558, 861, 690, 828, 759, 1035, 744, 1113, 870, 1104, 972, 1326, 945, 1380, 1116, 1386, 1218

Offset: 1

N. J. A. Sloane

			From _Gus Wiseman_, Oct 14 2020: (Start)
The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,2,4)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,3,3)  (1,3,4)
                    (2,1,2)  (1,4,1)  (1,4,2)  (1,4,3)
                    (2,2,1)  (2,1,3)  (1,5,1)  (1,5,2)
                    (3,1,1)  (2,3,1)  (2,1,4)  (1,6,1)
                             (3,1,2)  (2,2,3)  (2,1,5)
                             (3,2,1)  (2,3,2)  (2,3,3)
                             (4,1,1)  (2,4,1)  (2,5,1)
                                      (3,1,3)  (3,1,4)
                                      (3,2,2)  (3,2,3)
                                      (3,3,1)  (3,3,2)
                                      (4,1,2)  (3,4,1)
                                      (4,2,1)  (4,1,3)
                                      (5,1,1)  (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
N. J. A. Sloane, Transforms

A000010 is the length-2 version.
A000217(n-2) does not require relative primality.
A000740 counts these compositions of any length.
A000742 is the length-4 version.
A000837 counts relatively prime partitions.
A023023 is the unordered version.
A101271 is the strict case.
A101391 has this as column k = 3.
A284825*6 is the pairwise non-coprime case.
A291166 intersected with A014311 ranks these compositions.
A337461 is the pairwise coprime instead of relatively prime version.
A337603 counts length-3 compositions whose distinct parts are pairwise coprime.
A337604 is the pairwise non-coprime instead of relatively prime version.
Cf. A001399, A007997, A023022, A078374, A337450, A337451, A337602.

with(numtheory):
mobtr:= proc(p)
          proc(n) option remember;
            add(mobius(n/d)*p(d), d=divisors(n))
          end
        end:
A000217:= n-> n*(n+1)/2:
a:= mobtr(n-> A000217(n-2)):
seq(a(n), n=1..58);  # Alois P. Heinz, Feb 08 2011

mobtr[p_] := Module[{f}, f[n_] := f[n] = Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; f]; A000217[n_] := n*(n+1)/2; a = mobtr[A000217[#-2]&]; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Mar 12 2014, after Alois P. Heinz *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],GCD@@#==1&]],{n,0,30}] (* Gus Wiseman, Oct 14 2020 *)

Moebius transform of A000217(n-2).

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (1 - 3*x + 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 26 2017

Edited by Alois P. Heinz, Feb 08 2011

cp's OEIS Frontend

A007304 Sphenic numbers: products of 3 distinct primes.

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A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

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A307719 Number of partitions of n into 3 mutually coprime parts.

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A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

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A023023 Number of partitions of n into 3 unordered relatively prime parts.

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

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