A078374 -id:A078374 - cp's OEIS frontend

A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

1, 2, 6, 26, 78, 202, 606, 794, 2382, 2462, 2626, 7386, 7878, 8914, 10322, 12178, 26742, 30966, 32006, 36534, 42374, 43954, 47206, 80194, 96018, 115882, 127122, 131862, 141618, 149782, 158314, 160978, 184622, 217058, 240582, 248662, 260422, 347646, 449346

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1 or its prime indices are distinct, relatively prime, and already belong to the sequence.

			78 = prime(1)*prime(2)*prime(6) belongs to the sequence because the indices {1,2,6} are relatively prime, distinct, and already belong to the sequence.
The sequence of all identity RPMG-trees preceded by their Matula-Goebel numbers begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
    78: (o(o)(o(o)))
   202: (o(o(o(o))))
   606: (o(o)(o(o(o))))
   794: (o(o(o)(o(o))))
  2382: (o(o)(o(o)(o(o))))
  2462: (o(o(o(o(o)))))
  2626: (o(o(o))(o(o(o))))
  7386: (o(o)(o(o(o(o)))))
  7878: (o(o)(o(o))(o(o(o))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302696, A302796, A316467, A316470, A316471, A316474, A316494.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#==1,And[SquareFreeQ[#],GCD@@primeMS[#]==1,And@@#0/@primeMS[#]]]&]

A316503 Matula-Goebel numbers of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 41, 47, 55, 58, 62, 66, 78, 79, 82, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 143, 145, 155, 158, 165, 174, 179, 186, 195, 202, 205, 211, 218, 226, 246, 254, 257, 271, 274, 282, 286, 290, 293

Offset: 1

Gus Wiseman, Jul 05 2018

nonn

			Sequence of rooted identity trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: (o(o))
  10: (o((o)))
  11: ((((o))))
  13: ((o(o)))
  15: ((o)((o)))
  22: (o(((o))))
  26: (o(o(o)))
  29: ((o((o))))
  30: (o(o)((o)))
  31: (((((o)))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302796, A316467, A316470, A316494, A316502.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,And[SquareFreeQ[#],Or[PrimeQ[#],GCD@@primeMS[#]==1],And@@#0/@primeMS[#]]]&]

A319301 Sum of GCDs of strict integer partitions of n.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 11, 14, 18, 21, 22, 33, 30, 39, 49, 54, 54, 78, 72, 100, 110, 121, 126, 181, 174, 207, 238, 284, 284, 389, 370, 466, 512, 582, 647, 806, 796, 954, 1066, 1265, 1300, 1616, 1652, 1979, 2192, 2452, 2636, 3202, 3336, 3892, 4237, 4843, 5172, 6090

Offset: 1

Gus Wiseman, Sep 16 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Wikipedia, Partition (number theory)

Cf. A000009, A000010, A078374, A078392, A289508, A289509, A303138, A306956, A319300.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 `if`(i b(n$2, 0):
seq(a(n), n=1..61);  # Alois P. Heinz, Mar 17 2019

Table[Sum[GCD@@ptn,{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]
(* Second program: *)
b[n_, i_, r_] := b[n, i, r] = If[i(i+1)/2 < n, 0,
     With[{t = GCD[i, r]}, If[i < n, b[n - i, Min[i - 1, n - i], t], 0] +
     If[i == n, t, 0] + b[n, i - 1, r]]];
a[n_] := b[n, n, 0];
Array[a, 61] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

From Richard L. Ollerton, May 06 2021: (Start)
a(n) = Sum_{d|n} A000010(n/d)*A000009(d).
a(n) = Sum_{k=1..n} A000009(gcd(n,k)).
a(n) = Sum_{k=1..n} A000009(n/gcd(n,k))*A000010(gcd(n,k))/A000010(n/gcd(n,k)). (End)

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}]

A324752 Number of strict integer partitions of n not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 1, 4, 4, 4, 5, 6, 7, 10, 9, 12, 12, 16, 17, 22, 22, 26, 31, 35, 37, 46, 50, 55, 66, 70, 82, 90, 101, 114, 127, 143, 159, 172, 202, 215, 246, 267, 301, 327, 366, 402, 447, 491, 545, 600, 655, 722, 795, 875, 964, 1050, 1152, 1259, 1383

Offset: 0

Gus Wiseman, Mar 16 2019

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.

			The a(2) = 1 through a(17) = 12 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8  9   A   B    C   D    E    F    G    H
              42  43     54  64  65   75  76   86   87   97   98
                  52     63  73  83   84  85   95   96   A6   A7
                         72  82  542  93  94   A4   A5   C4   B6
                                      A2  B2   B3   B4   D3   C5
                                          643  752  C3   E2   D4
                                               842  D2   763  E3
                                                    654  943  854
                                                    843  A42  863
                                                    852       872
                                                              A52
                                                              B42
An example for n = 60 is (19,14,13,7,5,2), with prime indices:
  19: {8}
  14: {1,4}
  13: {6}
   7: {4}
   5: {3}
   2: {1}
None of these prime indices {1,3,4,6,8} belong to the partition, as required.

The subset version is A324742, with maximal case is A324763. The non-strict version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324743, A324748, A324751, A324756, A324758.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{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&]

A338553 Number of integer partitions of n that are either constant or relatively prime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 20, 29, 37, 56, 68, 101, 122, 170, 213, 297, 352, 490, 587, 778, 948, 1255, 1488, 1953, 2337, 2983, 3585, 4565, 5393, 6842, 8123, 10088, 12015, 14865, 17534, 21637, 25527, 31085, 36701, 44583, 52262, 63261, 74175, 88936, 104305, 124754

Offset: 0

Gus Wiseman, Nov 03 2020

nonn

The Heinz numbers of these partitions are given by A338555 = A000961 \/ A289509. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The a(1) = 1 through a(7) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (51)      (52)
                    (211)   (221)    (222)     (61)
                    (1111)  (311)    (321)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

A023022(n) + A059841(n) is the 2-part version.
A078374(n) + 1 is the strict case (n > 1).
A338554 counts the complement, with Heinz numbers A338552.
A338555 gives the Heinz numbers of these partitions.
A000005 counts constant partitions, with Heinz numbers A000961.
A000837 counts relatively prime partitions, with Heinz numbers A289509.
A282750 counts relatively prime partitions by sum and length.
Cf. A000010, A007360, A008284, A023023, A051424, A101271, A101391, A302698, A304712, A327516, A337664.

Table[Length[Select[IntegerPartitions[n],SameQ@@#||GCD@@#==1&]],{n,0,30}]

For n > 0, a(n) = A000005(n) + A000837(n) - 1.

A366853 Number of integer partitions of n into odd, pairwise coprime parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 18, 20, 22, 25, 29, 33, 36, 39, 43, 49, 55, 61, 66, 69, 75, 85, 94, 104, 113, 120, 129, 143, 159, 172, 183, 193, 207, 226, 251, 272, 288, 304, 325, 350, 383, 414, 437, 460, 494, 532, 577, 622, 655, 684

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(1) = 1 through a(10) = 7 partitions:
1  11  3    31    5      51      7        53        9          73
       111  1111  311    3111    511      71        531        91
                  11111  111111  31111    5111      711        5311
                                 1111111  311111    51111      7111
                                          11111111  3111111    511111
                                                    111111111  31111111
                                                               1111111111

Partitions into odd parts are counted by A000009, ranks A066208.
Allowing even parts gives A051424.
For relatively prime (not pairwise coprime): A366843, with evens A000837.
A000041 counts integer partitions, strict A000009 (also into odds).
A101268 counts pairwise coprime compositions.
A168532 counts partitions by gcd.
Cf. A007359, A018783, A055922, A078374, A337485, A366844, A366852.

pwcop[y_]:=And@@(GCD@@#==1&)/@Subsets[y,{2}]
Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&pwcop[#]&]],{n,0,30}]

A318718 Heinz numbers of strict integer partitions with a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 173, 179, 181, 183, 185, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Cf. A000009, A000837, A051424, A078374, A289509, A302696, A302796, A302797, A303280, A305713.

Select[Range[200],And[SquareFreeQ[#],GCD@@PrimePi/@FactorInteger[#][[All,1]]>1]&]

cp's OEIS Frontend

A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

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A316503 Matula-Goebel numbers of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

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A319301 Sum of GCDs of strict integer partitions of n.

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

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A324752 Number of strict integer partitions of n not containing 1 or any prime indices of the parts.

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Mathematica

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

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

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A338553 Number of integer partitions of n that are either constant or relatively prime.

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A366853 Number of integer partitions of n into odd, pairwise coprime parts.

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