A337605 -id:A337605 - cp's OEIS frontend

A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 4, 3, 5, 0, 9, 0, 9, 5, 10, 0, 16, 2, 14, 7, 17, 0, 27, 1, 21, 11, 24, 6, 36, 1, 30, 15, 37, 2, 51, 1, 41, 25, 44, 2, 64, 5, 58, 25, 57, 2, 81, 13, 69, 31, 70, 3, 108, 5, 80, 43, 85, 17, 123, 5, 97, 46, 120, 6, 144, 6

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

First differs from A082024 at a(31) = 1, A082024(31) = 0.

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

			The a(6) = 1 through a(16) = 5 partitions are (empty columns indicated by dots, A..G = 10..16):
  222  .  422  333  442  .  444  .  644  555  664  .  666  .  866
                    622     633     662  663  844     864     884
                            642     842  933  862     882     A55
                            822     A22       A42     963     A64
                                              C22     A44     A82
                                                      A62     C44
                                                      C33     C62
                                                      C42     E42
                                                      E22     G22

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

A014612 intersected with A337694 ranks these partitions.
A200976 and A328673 count these partitions of any length.
A284825 is the case that is also relatively prime.
A307719 is the pairwise coprime instead of non-coprime version.
A335402 gives the positions of zeros.
A337604 is the ordered version.
A337605 is the strict case.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts strict pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000212, A000217, A001840, A018783, A082024, A211540, A220377, A337461.

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

A337667 Number of compositions of n where any two parts have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 5, 1, 8, 4, 17, 1, 38, 1, 65, 19, 128, 1, 284, 1, 518, 67, 1025, 1, 2168, 16, 4097, 256, 8198, 1, 16907, 7, 32768, 1027, 65537, 79, 133088, 19, 262145, 4099, 524408, 25, 1056731, 51, 2097158, 16636, 4194317, 79, 8421248, 196, 16777712

Offset: 0

Gus Wiseman, Oct 05 2020

nonn

First differs from A178472 at a(31) = 7, a(31) = 1.

			The a(2) = 1 through a(10) = 17 compositions (A = 10):
   2   3   4    5   6     7   8      9     A
           22       24        26     36    28
                    33        44     63    46
                    42        62     333   55
                    222       224          64
                              242          82
                              422          226
                              2222         244
                                           262
                                           424
                                           442
                                           622
                                           2224
                                           2242
                                           2422
                                           4222
                                           22222

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..290

A101268 = 1 + A337462 is the pairwise coprime version.
A328673 = A200976 + 1 is the unordered version.
A337604 counts these compositions of length 3.
A337666 ranks these compositions.
A337694 gives Heinz numbers of the unordered version.
A337983 is the strict case.
A051185 counts intersecting set-systems, with spanning case A305843.
A318717 is the unordered strict case.
A319786 is the version for factorizations, with strict case A318749.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A051424, A082024, A178472, A284825, A319752, A327039, A337599, A337605, A337665.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],stabQ[#,CoprimeQ]&]],{n,0,15}]

A337694 Numbers with no two relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 179, 181, 183, 185, 189, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 23 2020

nonn

First differs from A305078 in having 1 and lacking 195.
First differs from A305103 in having 1 and 169 and lacking 195.
First differs from A328336 in lacking 897, with prime indices (2,6,9).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also Heinz numbers of integer partitions in which no two parts are relatively prime. 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:
   1: {}      37: {12}     79: {22}      121: {5,5}
   3: {2}     39: {2,6}    81: {2,2,2,2} 125: {3,3,3}
   5: {3}     41: {13}     83: {23}      127: {31}
   7: {4}     43: {14}     87: {2,10}    129: {2,14}
   9: {2,2}   47: {15}     89: {24}      131: {32}
  11: {5}     49: {4,4}    91: {4,6}     133: {4,8}
  13: {6}     53: {16}     97: {25}      137: {33}
  17: {7}     57: {2,8}   101: {26}      139: {34}
  19: {8}     59: {17}    103: {27}      147: {2,4,4}
  21: {2,4}   61: {18}    107: {28}      149: {35}
  23: {9}     63: {2,2,4} 109: {29}      151: {36}
  25: {3,3}   65: {3,6}   111: {2,12}    157: {37}
  27: {2,2,2} 67: {19}    113: {30}      159: {2,16}
  29: {10}    71: {20}    115: {3,9}     163: {38}
  31: {11}    73: {21}    117: {2,2,6}   167: {39}

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

A200976 and A328673 count these partitions.
A302696 and A302569 are pairwise coprime instead of pairwise non-coprime.
A318719 is the squarefree case.
A328867 looks at distinct prime indices.
A337666 is the version for standard compositions.
A101268 counts pairwise coprime or singleton compositions.
A318717 counts strict pairwise non-coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337667 counts pairwise non-coprime compositions.
Cf. A051185, A051424, A056239, A112798, A220377, A284825, A302797, A303282, A305843, A319752, A336737, A337599, A337604, A337605.

filter:= proc(n) local F,i,j,np;
  if n::even and n>2 then return false fi;
  F:= map(t -> numtheory:-pi(t[1]), ifactors(n)[2]);
  np:= nops(F);
  for i from 1 to np-1 do
    for j from i+1 to np do
      if igcd(F[i],F[j])=1 then return false fi
  od od;
  true
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 06 2020

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Select[Range[100],stabQ[primeMS[#],CoprimeQ]&]

A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 128, 130, 136, 138, 160, 162, 168, 170, 256, 260, 288, 292, 512, 514, 520, 522, 528, 544, 546, 552, 554, 640, 642, 648, 650, 672, 674, 680, 682, 1024, 2048, 2050, 2052, 2056, 2058, 2080, 2082, 2084, 2088, 2090, 2176

Offset: 1

Gus Wiseman, Oct 05 2020

nonn

Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).
This is a ranking sequence for pairwise non-coprime compositions.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
       0: ()          138: (4,2,2)       546: (4,4,2)
       2: (2)         160: (2,6)         552: (4,2,4)
       4: (3)         162: (2,4,2)       554: (4,2,2,2)
       8: (4)         168: (2,2,4)       640: (2,8)
      10: (2,2)       170: (2,2,2,2)     642: (2,6,2)
      16: (5)         256: (9)           648: (2,4,4)
      32: (6)         260: (6,3)         650: (2,4,2,2)
      34: (4,2)       288: (3,6)         672: (2,2,6)
      36: (3,3)       292: (3,3,3)       674: (2,2,4,2)
      40: (2,4)       512: (10)          680: (2,2,2,4)
      42: (2,2,2)     514: (8,2)         682: (2,2,2,2,2)
      64: (7)         520: (6,4)        1024: (11)
     128: (8)         522: (6,2,2)      2048: (12)
     130: (6,2)       528: (5,5)        2050: (10,2)
     136: (4,4)       544: (4,6)        2052: (9,3)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

A337604 counts these compositions of length 3.
A337667 counts these compositions.
A337694 is the version for Heinz numbers of partitions.
A337696 is the strict case.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 count pairwise non-coprime partitions.
A318717 counts strict pairwise non-coprime partitions.
A327516 counts pairwise coprime partitions.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
All of the following pertain to compositions in standard order (A066099):
- A000120 is length.
- A070939 is sum.
- A124767 counts runs.
- A233564 ranks strict compositions.
- A272919 ranks constant compositions.
- A291166 appears to rank relatively prime compositions.
- A326674 is greatest common divisor.
- A333219 is Heinz number.
- A333227 ranks coprime (Mathematica definition) compositions.
- A333228 ranks compositions with distinct parts coprime.
- A335235 ranks singleton or coprime compositions.
Cf. A082024, A284825, A305713, A319752, A319786, A327039, A327040, A336737, A337599, A337605.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Select[Range[0,1000],stabQ[stc[#],CoprimeQ]&]

A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 7, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 7, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 13, 0, 1, 2, 0, 1, 7, 0, 2, 1, 7, 0, 6, 0, 1, 2, 2, 1, 7, 0, 4, 0, 1, 0, 13, 1, 1, 1, 3, 0, 13, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 7, 0

Offset: 1

K. B. Subramaniam (kb_subramaniambalu(AT)yahoo.com) and Amarnath Murthy, Dec 24 2001

nonn

a(m) = a(n) if m and n have same factorization structure.

			a(24) = 3: the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The triples are (1, 2, 3), (1, 2, 9), (1, 3, 4).
a(30) = 7: the triples are (1, 2, 3), (1, 2, 5), (1, 3, 5), (2, 3, 5), (1, 3, 10), (1, 5, 6), (1, 2, 15).

Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, vol. 12, No. 1-2-3, Spring 2001.pp 303-306.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positions of zeros are A000961.
Positions of ones are A006881.
The version for subsets of {1..n} instead of divisors is A015617.
The non-strict ordered version is A048785.
The version for pairs of divisors is A063647.
The non-strict version (3-multisets) is A100565.
The version for partitions is A220377 (non-strict: A307719).
A version for sets of divisors of any size is A225520.
A000005 counts divisors.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A018892 counts unordered pairs of coprime divisors (ordered: A048691).
A051026 counts pairwise indivisible subsets of {1..n}.
A337461 counts 3-part pairwise coprime compositions.
A338331 lists Heinz numbers of pairwise coprime partitions.
Cf. A007360, A023022, A084422, A276187, A282935, A305713, A337563, A337605, A343652, A343655.

Table[Length[Select[Subsets[Divisors[n],{3}],CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Apr 28 2021 *)

A066620(n) = (numdiv(n^3)-3*numdiv(n)+2)/6; \\ After Jovovic's formula. - Antti Karttunen, May 27 2017

from sympy import divisor_count as d
def a(n): return (d(n**3) - 3*d(n) + 2)/6 # Indranil Ghosh, May 27 2017

In the reference it is shown that if k is a squarefree number with r prime factors and m with (r+1) prime factors then a(m) = 4*a(k) + 2^k - 1.

a(n) = (tau(n^3)-3*tau(n)+2)/6. - Vladeta Jovovic, Nov 27 2004

More terms from Vladeta Jovovic, Apr 03 2003
Name corrected by Andrey Zabolotskiy, Dec 09 2020
Name corrected by Gus Wiseman, Apr 28 2021 (ordered version is 6*a(n))

A337482 Number of compositions of n that are neither strictly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 2, 7, 18, 45, 101, 219, 461, 957, 1957, 3978, 8036, 16182, 32506, 65202, 130642, 261601, 523598, 1047709, 2096062, 4192946, 8386912, 16775117, 33551832, 67105663, 134213789, 268430636, 536865013, 1073734643, 2147474910, 4294956706, 8589921771

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(4) = 2 through a(4) = 18 compositions:
  (112)  (113)   (114)
  (121)  (122)   (132)
         (131)   (141)
         (212)   (213)
         (1112)  (231)
         (1121)  (312)
         (1211)  (1113)
                 (1122)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11112)
                 (11121)
                 (11211)
                 (12111)

Ranked by the complement of the intersection of A114994 and A333255.
A128422 counts only the case of length 3.
A218004 counts the complement.
A332834 is the weak version.
A337481 is the strict version.
A001523 counts unimodal compositions, with complement counted by A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A332745/A332835 count partitions/compositions with weakly increasing or weakly decreasing run-lengths.
Cf. A216652, A329398, A332831, A332833, A337462, A337483, A337484, A337605.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!GreaterEqual@@#&]],{n,0,15}]

a(n) = 2^(n-1) - A000009(n) - A000041(n) + 1, n > 0.

A338557 Products of three distinct prime numbers of even index.

Original entry on oeis.org

273, 399, 609, 741, 777, 903, 1113, 1131, 1281, 1443, 1491, 1653, 1659, 1677, 1729, 1869, 2067, 2109, 2121, 2247, 2373, 2379, 2451, 2639, 2751, 2769, 2919, 3021, 3081, 3171, 3219, 3367, 3423, 3471, 3477, 3633, 3741, 3801, 3857, 3913, 3939, 4047, 4053, 4173

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.
Also sphenic numbers (A007304) with all even prime indices (A031215).
Also Heinz numbers of strict integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     273: {2,4,6}     1869: {2,4,24}    3219: {2,10,12}
     399: {2,4,8}     2067: {2,6,16}    3367: {4,6,12}
     609: {2,4,10}    2109: {2,8,12}    3423: {2,4,38}
     741: {2,6,8}     2121: {2,4,26}    3471: {2,6,24}
     777: {2,4,12}    2247: {2,4,28}    3477: {2,8,18}
     903: {2,4,14}    2373: {2,4,30}    3633: {2,4,40}
    1113: {2,4,16}    2379: {2,6,18}    3741: {2,10,14}
    1131: {2,6,10}    2451: {2,8,14}    3801: {2,4,42}
    1281: {2,4,18}    2639: {4,6,10}    3857: {4,8,10}
    1443: {2,6,12}    2751: {2,4,32}    3913: {4,6,14}
    1491: {2,4,20}    2769: {2,6,20}    3939: {2,6,26}
    1653: {2,8,10}    2919: {2,4,34}    4047: {2,8,20}
    1659: {2,4,22}    3021: {2,8,16}    4053: {2,4,44}
    1677: {2,6,14}    3081: {2,6,22}    4173: {2,6,28}
    1729: {4,6,8}     3171: {2,4,36}    4179: {2,4,46}

For the following, NNS means "not necessarily strict".
A007304 allows all prime indices (not just even) (NNS: A014612).
A046389 allows all odd primes (NNS: A046316).
A258117 allows products of any length (NNS: A066207).
A307534 is the version for odds instead of evens (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338556 is the NNS version.
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers, with even case A039956.
A078374 counts 3-part relatively prime strict partitions (NNS: A023023).
A075819 lists even Heinz numbers of strict triples (NNS: A075818).
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
A258116 lists squarefree numbers with all odd prime indices (NNS: A066208).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337605.

Select[Range[1000],SquareFreeQ[#]&&PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (omega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, nextprime, integer_nthroot
def A338557(n):
    def bisection(f,kmin=0,kmax=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
    def f(x): return int(n+x-sum((primepi(x//(k*m))>>1)-(b>>1) for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 25, 55, 117, 241, 493, 1001, 2019, 4061, 8149, 16331, 32705, 65461, 130981, 262037, 524161, 1048425, 2096975, 4194097, 8388365, 16776933, 33554103, 67108481, 134217285, 268434945, 536870321, 1073741145, 2147482869, 4294966401, 8589933569

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(2) = 1 through a(5) = 11 compositions:
  (11)  (111)  (22)    (113)
               (112)   (122)
               (121)   (131)
               (211)   (212)
               (1111)  (221)
                       (311)
                       (1112)
                       (1121)
                       (1211)
                       (2111)
                       (11111)

Ranked by the complement of the intersection of A333255 and A333256.
A332834 is the weak version.
A337482 is the semi-strict version.
A337484 counts only compositions of length 3.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A218004 counts strictly increasing or weakly decreasing compositions.
Cf. A216652, A329398, A337462, A337483, A337605.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!Greater@@#&]],{n,0,15}]

a(n) = 2^(n-1) - 2*A000009(n) + 1, n > 0.

A337983 Number of compositions of n into distinct parts, any two of which have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 5, 1, 13, 1, 13, 7, 19, 1, 35, 1, 59, 15, 65, 1, 117, 5, 133, 27, 195, 1, 411, 7, 435, 67, 617, 17, 941, 7, 1177, 135, 1571, 13, 2939, 31, 3299, 375, 4757, 13, 6709, 43, 8813, 643, 11307, 61, 16427, 123, 24331, 1203, 30461, 67

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

Number of pairwise non-coprime strict compositions of n.

			The a(2) = 1 through a(15) = 7 compositions (A..F = 10..15):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F
              24     26  36  28     2A      2C   3C
              42     62  63  46     39      4A   5A
                             64     48      68   69
                             82     84      86   96
                                    93      A4   A5
                                    A2      C2   C3
                                    246     248
                                    264     284
                                    426     428
                                    462     482
                                    624     824
                                    642     842

A318717 is the unordered version.
A318719 is the version for Heinz numbers of partitions.
A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
A337605*6 counts these compositions of length 3.
A337667 is the non-strict version, ranked by A337666.
A337696 ranks these compositions.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 are the unordered version.
A233564 ranks strict compositions.
A318749 is the version for factorizations, with non-strict version A319786.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337694 lists numbers with no two relatively prime prime indices.
Cf. A082024, A284825, A302797, A305713, A319752, A327516, A333227, A335235, A336737, A337604.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,30}]

A338333 Number of relatively prime 3-part strict integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 7, 6, 10, 8, 14, 12, 18, 16, 24, 18, 30, 25, 34, 30, 44, 31, 52, 42, 56, 49, 69, 50, 80, 64, 83, 70, 102, 71, 114, 90, 112, 100, 140, 98, 153, 117, 153, 132, 184, 128, 195, 154, 196, 169, 234, 156, 252, 196, 241

Offset: 0

Gus Wiseman, Oct 30 2020

nonn

The Heinz numbers of these partitions are the intersection of A005117 (strict), A005408 (no 1's), A014612 (length 3), and A289509 (relatively prime).

			The a(9) = 1 through a(19) = 14 triples (A = 10, B = 11, C = 12, D = 13, E = 14):
  432   532   542   543   643   653   654   754   764   765   865
              632   732   652   743   753   763   854   873   874
                          742   752   762   853   863   954   964
                          832   932   843   943   872   972   973
                                      852   952   953   A53   982
                                      942   B32   962   B43   A54
                                      A32         A43   B52   A63
                                                  A52   D32   A72
                                                  B42         B53
                                                  C32         B62
                                                              C43
                                                              C52
                                                              D42
                                                              E32

A001399(n-9) does not require relative primality.
A005117 /\ A005408 /\ A014612 /\ A289509 gives the Heinz numbers.
A055684 is the 2-part version.
A284825 counts the case that is also pairwise non-coprime.
A337452 counts these partitions of any length.
A337563 is the pairwise coprime instead of relatively prime version.
A337605 is the pairwise non-coprime instead of relative prime version.
A338332 is the not necessarily strict version.
A338333*6 is the ordered version.
A000837 counts relatively prime partitions.
A008284 counts partitions by sum and length.
A078374 counts relatively prime strict partitions.
A101271 counts 3-part relatively prime strict partitions.
A220377 counts 3-part pairwise coprime strict partitions.
A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
Cf. A000010, A000217, A000741, A023022, A082024, A302698, A307719, A337450, A337599, A338468.

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

cp's OEIS Frontend

A337599 Number of unordered triples of positive integers summing to n, any two of which have a common divisor > 1.

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A337667 Number of compositions of n where any two parts have a common divisor > 1.

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A337694 Numbers with no two relatively prime prime indices.

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A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

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A066620 Number of unordered triples of distinct pairwise coprime divisors of n.

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A337482 Number of compositions of n that are neither strictly increasing nor weakly decreasing.

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A338557 Products of three distinct prime numbers of even index.

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A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

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