A337450 -id:A337450 - cp's OEIS frontend

A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

0, 0, 0, 0, 1, 0, 3, 2, 5, 4, 13, 7, 23, 18, 32, 33, 65, 50, 104, 92, 148, 153, 252, 226, 376, 376, 544, 570, 846, 821, 1237, 1276, 1736, 1869, 2552, 2643, 3659, 3887, 5067, 5509, 7244, 7672, 10086, 10909, 13756, 15168, 19195, 20735, 26237, 28708, 35418, 39207

Offset: 1

Gus Wiseman, Apr 11 2018

nonn

Two or more numbers are relatively prime if they have no common divisor other than 1. A single number is not considered relatively prime unless it is equal to 1 (which is impossible in this case).

The Heinz numbers of these partitions are given by A302697.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (73)    (65)     (75)
           (52)   (332)  (72)    (433)   (74)     (543)
           (322)         (432)   (532)   (83)     (552)
                         (522)   (3322)  (92)     (732)
                         (3222)          (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

A000837 is the version allowing 1's.
A002865 does not require relative primality.
A302697 gives the Heinz numbers of these partitions.
A337450 is the ordered version.
A337451 is the ordered strict version.
A337452 is the strict version.
A337485 is the pairwise coprime instead of relatively prime version.
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A332004 counts strict relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A338332 is the case of length 3, with strict case A338333.
Cf. A007359, A018783, A051424, A101268, A289508, A289509, A302568, A337563, A337984, A338468.

b:= proc(n, i, g) option remember; `if`(n=0, `if`(g=1, 1, 0),
      `if`(i<2, 0, b(n, i-1, g)+b(n-i, min(n-i, i), igcd(g, i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 12 2018

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&GCD@@#===1&]],{n,30}]
(* Second program: *)
b[n_, i_, g_] := b[n, i, g] = If[n == 0, If[g == 1, 1, 0], If[i < 2, 0, b[n, i - 1, g] + b[n - i, Min[n - i, i], GCD[g, i]]]];
a[n_] := b[n, n, 0];
Array[a, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = A002865(n) - A018783(n).

Extended by Gus Wiseman, Oct 29 2020

A337485 Number of pairwise coprime integer partitions of n with no 1's, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 3, 5, 4, 4, 7, 8, 9, 10, 10, 9, 13, 17, 18, 17, 19, 19, 24, 29, 34, 33, 31, 31, 42, 42, 56, 55, 50, 54, 66, 77, 86, 86, 79, 81, 96, 124, 127, 126, 127, 126, 145, 181, 190, 184, 183, 192, 212, 262, 289, 278, 257, 270, 311

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

Such a partition is necessarily strict.

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A005117 (strict), and A302696 (coprime).

			The a(n) partitions for n = 5, 7, 12, 13, 16, 17, 18, 19 (A..H = 10..17):
  (3,2)  (4,3)  (7,5)    (7,6)  (9,7)    (9,8)      (B,7)    (A,9)
         (5,2)  (5,4,3)  (8,5)  (B,5)    (A,7)      (D,5)    (B,8)
                (7,3,2)  (9,4)  (D,3)    (B,6)      (7,6,5)  (C,7)
                         (A,3)  (7,5,4)  (C,5)      (8,7,3)  (D,6)
                         (B,2)  (8,5,3)  (D,4)      (9,5,4)  (E,5)
                                (9,5,2)  (E,3)      (9,7,2)  (F,4)
                                (B,3,2)  (F,2)      (B,4,3)  (G,3)
                                         (7,5,3,2)  (B,5,2)  (H,2)
                                                    (D,3,2)  (B,5,3)
                                                             (7,5,4,3)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750

A005408 intersected with A302696 ranks these partitions.
A007359 considers all singletons to be coprime.
A327516 allows 1's, with non-strict version A305713.
A337452 is the relatively prime instead of pairwise coprime version, with non-strict version A302698.
A337563 is the restriction to partitions of length 3.
A002865 counts partitions with no 1's.
A078374 counts relatively prime strict partitions.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A101268, A220377, A302696, A304709, A332004, A337450, A337451, A337462, A337561, A337605.

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

a(n) = A007359(n) - 1 for n > 1.

A055684 Number of different n-pointed stars.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 1, 4, 1, 5, 2, 3, 3, 7, 2, 8, 3, 5, 4, 10, 3, 9, 5, 8, 5, 13, 3, 14, 7, 9, 7, 11, 5, 17, 8, 11, 7, 19, 5, 20, 9, 11, 10, 22, 7, 20, 9, 15, 11, 25, 8, 19, 11, 17, 13, 28, 7, 29, 14, 17, 15, 23, 9, 32, 15, 21, 11, 34, 11, 35, 17, 19, 17, 29, 11

Offset: 3

Robert G. Wilson v, Jun 09 2000

Does not count rotations or reflections.
This is also the distinct ways of writing a number as the sum of two positive integers greater than one that are coprimes. - Lei Zhou, Mar 19 2014
Equivalently, a(n) is the number of relatively prime 2-part partitions of n without 1's. The Heinz numbers of these partitions are the intersection of A001358 (pairs), A005408 (no 1's), and A000837 (relatively prime) or A302696 (pairwise coprime). - Gus Wiseman, Oct 28 2020

			The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties.
From _Gus Wiseman_, Oct 28 2020: (Start)
The a(5) = 1 through a(17) = 7 irreducible pairs > 1 (shown as fractions, empty column indicated by dot):
  2/3  .  2/5  3/5  2/7  3/7  2/9  5/7  2/11  3/11  2/13  3/13  2/15
          3/4       4/5       3/8       3/10  5/9   4/11  5/11  3/14
                              4/7       4/9         7/8   7/9   4/13
                              5/6       5/8                     5/12
                                        6/7                     6/11
                                                                7/10
                                                                8/9
(End)

Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 58.

Lei Zhou, Table of n, a(n) for n = 3..10002
Alexander Bogomolny, Polygons: formality and intuition.. Includes applet to draw star polygons.
Vi Hart, Doodling in Math Class: Stars, Video (2010).
Hugo Pfoertner, Star-shaped regular polygons up to n=25.
Eric Weisstein's World of Mathematics, Star Polygon

Cf. A023022.
Cf. A053669 smallest skip increment, A102302 skip increment of densest star polygon.
A055684*2 is the ordered version.
A082023 counts the complement (reducible pairs > 1).
A220377, A337563, and A338332 count triples instead of pairs.
A000837 counts relatively prime partitions, with strict case A078374.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A302698 counts relatively prime partitions with no 1's, with strict case A337452.
A327516 counts pairwise coprime partitions, with strict case A305713.
A337450 counts relatively prime compositions with no 1's, with strict case A337451.
Cf. A001399, A101268, A140106, A337461, A337462, A338333.

with(numtheory): A055684 := n->(phi(n)-2)/2; seq(A055684(n), n=3..100);

Table[(EulerPhi[n]-2)/2, {n, 3, 50}]
Table[Length[Select[IntegerPartitions[n,{2}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 28 2020 *)

a(n) = A023022(n) - 1.

a(n) + A082023(n) = A140106(n). - Gus Wiseman, Oct 28 2020

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

A337452 Number of relatively prime strict integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 2, 6, 3, 9, 7, 11, 11, 20, 15, 28, 24, 35, 36, 55, 47, 73, 71, 95, 96, 136, 123, 180, 177, 226, 235, 305, 299, 403, 406, 503, 523, 668, 662, 852, 873, 1052, 1115, 1370, 1391, 1720, 1784, 2125, 2252, 2701, 2786, 3348, 3520, 4116

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

			The a(5) = 1 through a(16) = 11 partitions (A = 10, B = 11, C = 12, D = 13):
  32  43  53  54   73   65   75   76   95    87    97
      52      72   532  74   543  85   B3    B4    B5
              432       83   732  94   653   D2    D3
                        92        A3   743   654   754
                        542       B2   752   753   763
                        632       643  932   762   853
                                  652  5432  843   943
                                  742        852   952
                                  832        942   B32
                                             A32   6532
                                             6432  7432

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

A078374 is the version allowing 1's.
A302698 is the non-strict version.
A332004 is the ordered version allowing 1's.
A337450 is the ordered non-strict version.
A337451 is the ordered version.
A337485 is the pairwise coprime version.
A000837 counts relatively prime partitions.
A078374 counts relatively prime strict partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A101268, A289509, A337485, A337563.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A337451 Number of relatively prime strict compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 10, 8, 20, 14, 34, 52, 72, 90, 146, 172, 244, 390, 502, 680, 956, 1218, 1686, 2104, 3436, 4078, 5786, 7200, 10108, 12626, 17346, 20876, 32836, 38686, 53674, 67144, 91528, 113426, 152810, 189124, 245884, 343350, 428494, 552548, 719156

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

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

			The a(5) = 2 through a(10) = 8 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)    (3,7)
  (3,2)     (3,4)  (5,3)  (4,5)    (7,3)
            (4,3)         (5,4)    (2,3,5)
            (5,2)         (7,2)    (2,5,3)
                          (2,3,4)  (3,2,5)
                          (2,4,3)  (3,5,2)
                          (3,2,4)  (5,2,3)
                          (3,4,2)  (5,3,2)
                          (4,2,3)
                          (4,3,2)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

A032022 does not require relative primality.
A302698 is the unordered non-strict version.
A332004 is the version allowing 1's.
A337450 is the non-strict version.
A337452 is the unordered version.
A000837 counts relatively prime partitions.
A032020 counts strict compositions.
A078374 counts strict relatively prime partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
A337561 counts strict pairwise coprime compositions.
Cf. A000010, A007359, A101268, A178472, A216652, A289509, A337562, A337563.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 4, 8, 8, 14, 10, 16, 12, 30, 38, 46, 46, 48, 52, 62, 152, 96, 156, 112, 190, 256, 338, 420, 394, 326, 402, 734, 622, 1150, 802, 946, 898, 1730, 1946, 2524, 2200, 2328, 2308, 3356, 5816, 4772, 5350, 4890, 6282, 6316, 12092, 8902

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. These compositions must be strict.

			The a(5) = 2 through a(12) = 14 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)  (3,7)    (2,9)  (5,7)
  (3,2)     (3,4)  (5,3)  (4,5)  (7,3)    (3,8)  (7,5)
            (4,3)         (5,4)  (2,3,5)  (4,7)  (2,3,7)
            (5,2)         (7,2)  (2,5,3)  (5,6)  (2,7,3)
                                 (3,2,5)  (6,5)  (3,2,7)
                                 (3,5,2)  (7,4)  (3,4,5)
                                 (5,2,3)  (8,3)  (3,5,4)
                                 (5,3,2)  (9,2)  (3,7,2)
                                                 (4,3,5)
                                                 (4,5,3)
                                                 (5,3,4)
                                                 (5,4,3)
                                                 (7,2,3)
                                                 (7,3,2)

A022340 intersected with A333227 is a ranking sequence (using standard compositions A066099) for these compositions.
A212804 does not require coprimality, with unordered version A002865.
A337450 is the relatively prime instead of pairwise coprime version, with strict case A337451 and unordered version A302698.
A337462 allows 1's, with strict case A337561 (or A101268 with singletons), unordered version A327516 with Heinz numbers A302696, and 3-part case A337461.
A337485 is the unordered version (or A007359 with singletons considered coprime), with Heinz numbers A337984.
A337563 is the case of unordered triples.
Cf. A078374, A178472, A302568, A302697, A305713, A307719, A332004, A337562.

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

For n > 1, the version where singletons are considered coprime is a(n) + 1.

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}]

A338332 Number of relatively prime 3-part integer partitions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 3, 8, 6, 9, 9, 16, 10, 21, 15, 22, 20, 33, 21, 38, 30, 41, 35, 56, 34, 65, 49, 64, 56, 79, 55, 96, 72, 93, 77, 120, 76, 133, 99, 122, 110, 161, 105, 172, 126, 167, 143, 208, 136, 213, 165, 212, 182, 261, 163, 280, 210, 257

Offset: 0

Gus Wiseman, Oct 30 2020

nonn

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

			The a(7) = 1 through a(17) = 16 triples (A = 10, B = 11, C = 12, D = 13):
  322   332   432   433   443   543   544   554   654   655   665
              522   532   533   552   553   653   744   754   755
                          542   732   643   743   753   763   764
                          632         652   752   762   772   773
                          722         733   833   843   853   854
                                      742   932   852   943   863
                                      832         942   952   872
                                      922         A32   A33   944
                                                  B22   B32   953
                                                              962
                                                              A43
                                                              A52
                                                              B33
                                                              B42
                                                              C32
                                                              D22

A001399(n-6) does not require relative primality.
A005408 /\ A014612 /\ A289509 gives the Heinz numbers of these partitions.
A055684 is the 2-part version.
A284825 counts the case that is also pairwise non-coprime.
A302698 counts these partitions of any length.
A337563 is the pairwise coprime instead of relatively prime version.
A338333 is the strict version.
A000837 counts relatively prime partitions, with strict case A078374.
A008284 counts partitions by sum and length.
Cf. A000010, A000741, A023022, A078374, A082024, A101271, A307719, A337450, A337599, A337600, A337601.

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

A338468 Odd squarefree numbers whose prime indices have no common divisor > 1.

Original entry on oeis.org

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 205, 209, 215, 217, 219, 221, 231, 249, 253, 255, 265, 285, 287, 291, 295, 309, 323, 327, 329, 335, 341, 345, 355, 357, 381, 385, 391, 395, 403, 407, 411

Offset: 1

Gus Wiseman, Oct 29 2020

nonn

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 relatively prime strict integer partitions with no 1's (A337452). The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     15: {2,3}      145: {3,10}     249: {2,23}     355: {3,20}
     33: {2,5}      155: {3,11}     253: {5,9}      357: {2,4,7}
     35: {3,4}      161: {4,9}      255: {2,3,7}    381: {2,31}
     51: {2,7}      165: {2,3,5}    265: {3,16}     385: {3,4,5}
     55: {3,5}      177: {2,17}     285: {2,3,8}    391: {7,9}
     69: {2,9}      187: {5,7}      287: {4,13}     395: {3,22}
     77: {4,5}      195: {2,3,6}    291: {2,25}     403: {6,11}
     85: {3,7}      201: {2,19}     295: {3,17}     407: {5,12}
     93: {2,11}     205: {3,13}     309: {2,27}     411: {2,33}
     95: {3,8}      209: {5,8}      323: {7,8}      413: {4,17}
    105: {2,3,4}    215: {3,14}     327: {2,29}     415: {3,23}
    119: {4,7}      217: {4,11}     329: {4,15}     429: {2,5,6}
    123: {2,13}     219: {2,21}     335: {3,19}     435: {2,3,10}
    141: {2,15}     221: {6,7}      341: {5,11}     437: {8,9}
    143: {5,6}      231: {2,4,5}    345: {2,3,9}    447: {2,35}

A302568 is the prime or pairwise coprime version, counted by A007359.
A302697 is not required to be squarefree, counted by A302698 (ordered version: A337450).
A302796 allows evens, counted by A078374 (ordered version: A332004).
A337452 counts partitions with these Heinz numbers (ordered version: A337451).
A337984 is the pairwise coprime version, counted by A337485 (ordered version: A337697).
A005117 lists squarefree numbers.
A005408 lists odd numbers.
A056911 lists odd squarefree numbers.
A289509 lists Heinz numbers of relatively prime partitions, counted by A000837 (ordered version: A000740).
Cf. A000010, A007359, A051424, A055684, A056239, A101268, A289508, A302505, A302569, A302696, A302798, A337694.

Select[Range[1,100,2],SquareFreeQ[#]&&GCD@@PrimePi/@First/@FactorInteger[#]==1&]

cp's OEIS Frontend

A302698 Number of integer partitions of n into relatively prime parts that are all greater than 1.

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A337485 Number of pairwise coprime integer partitions of n with no 1's, where a singleton is not considered coprime unless it is (1).

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A055684 Number of different n-pointed stars.

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

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A337452 Number of relatively prime strict integer partitions of n with no 1's.

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A337451 Number of relatively prime strict compositions of n with no 1's.

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A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

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