A300486 -id:A300486 - cp's OEIS frontend

A302094 Number of relatively prime or monic twice-partitions of n.

1, 3, 6, 10, 27, 35, 113, 170, 396, 641, 1649, 2318, 5905, 9112, 18678, 32529, 69094, 106210, 227480, 363433, 705210, 1196190, 2325023, 3724233, 7192245, 11915884, 21857887, 36597843, 67406158, 109594872, 201747847, 333400746, 591125465, 987069077, 1743223350

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime). Then a relatively prime or monic twice-partition of n is a choice of a relatively prime or monic partition of each part in a relatively prime or monic partition of n.

			The a(4) = 10 relatively prime or monic twice-partitions:
(4), (31), (211), (1111),
(3)(1), (21)(1), (111)(1),
(2)(1)(1), (11)(1)(1),
(1)(1)(1)(1).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A300486, A301462, A301467, A301480, A302915, A302916, A302917.

ip[n_]:=ip[n]=Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&];
Table[Sum[Times@@Length/@ip/@ptn,{ptn,ip[n]}],{n,10}]

A302915 Number of relatively prime enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 4, 8, 28, 56, 256, 656, 2480, 6688, 30736, 73984, 366560, 1006720, 3966976, 12738560, 58427648, 148069632, 764473600, 2133585664, 8939502080, 28705390592, 136987259648, 356634376704, 1780025034240, 5455065263104, 23215437079552, 73123382895616

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime enriched p-tree of weight n is either a single node of weight n, or a finite sequence of two or more relatively prime enriched p-trees whose weights are weakly decreasing, relatively prime, and sum to n.

			The a(4) = 8 relatively prime enriched p-trees are 4, (31), ((21)1), (((11)1)1), ((111)1), (211), ((11)11), (1111). Missing from this list are the enriched p-trees ((11)(11)), ((11)2), (2(11)), (22).

Cf. A000081, A000837, A003238, A004111, A055277, A093637, A196545, A289501, A290689, A300486, A301462, A301467, A302094, A302916, A302917.

a[n_]:=a[n]=1+Sum[Times@@a/@y,{y,Rest[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]]}];
Array[a,20]

A302916 Number of relatively prime p-trees of weight n.

Original entry on oeis.org

1, 1, 2, 4, 11, 22, 74, 174, 530, 1302, 4713, 10639, 40877, 101795, 325609, 925733, 3432819, 8078511, 32542036, 82226383, 279096823, 795532677, 3066505569, 7374764180, 28946183035, 79313174765, 275507514909, 772692247626, 3049937788372, 7071057261148

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime p-tree of weight n is either a single node, or a finite sequence of two or more relatively prime p-trees whose weights are weakly decreasing, relatively prime, and sum to n.

			The a(4) = 4 relatively prime p-trees are (((oo)o)o), ((ooo)o), ((oo)oo), (oooo). Missing from this list is the p-tree ((oo)(oo)).

Cf. A000081, A000837, A003238, A004111, A055277, A093637, A196545, A289501, A290689, A300486, A301462, A301467, A302094, A302915, A302917.

a[n_]:=a[n]=If[n===1,1,Sum[Times@@a/@y,{y,Rest[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]]}]];
Array[a,20]

A302917 Solution to a(1) = 1 and Sum_y Product_i a(y_i) = 0 for each n > 1, where the sum is over all relatively prime or monic partitions of n.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -3, 1, 4, -5, -3, 3, 4, 2, -6, -6, 19, -8, -25, 25, 20, -12, -34, 2, 30, 38, -117, 54, 159, -173, -123, 55, 229, 32, -250, -148, 753, -365, -1022, 840, 1121, -847, -1482, -390, 2099

Offset: 1

Gus Wiseman, Apr 15 2018

sign

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

Cf. A000837, A036987, A047966, A063834, A093637, A196545, A220418, A289501, A290261, A300486, A300863-A300866, A301462, A302094, A302915, A302916.

a[n_]:=a[n]=If[n===1,1,0]-Sum[Times@@a/@y,{y,Rest[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]]}];
Array[a,20]

A320435 Regular triangle read by rows where T(n,k) is the number of relatively prime k-subsets of {1,...,n}, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 10, 5, 1, 1, 11, 19, 15, 6, 1, 1, 17, 34, 35, 21, 7, 1, 1, 21, 52, 69, 56, 28, 8, 1, 1, 27, 79, 125, 126, 84, 36, 9, 1, 1, 31, 109, 205, 251, 210, 120, 45, 10, 1, 1, 41, 154, 325, 461, 462, 330, 165, 55, 11, 1, 1, 45, 196

Offset: 1

Gus Wiseman, Jan 08 2019

Two or more numbers are relatively prime if they have no common divisor > 1. A single number is not considered to be relatively prime unless it is equal to 1.

			Triangle begins:
    1
    1    1
    1    3    1
    1    5    4    1
    1    9   10    5    1
    1   11   19   15    6    1
    1   17   34   35   21    7    1
    1   21   52   69   56   28    8    1
    1   27   79  125  126   84   36    9    1
    1   31  109  205  251  210  120   45   10    1
    1   41  154  325  461  462  330  165   55   11    1
    1   45  196  479  786  923  792  495  220   66   12    1
    1   57  262  699 1281 1715 1716 1287  715  286   78   13    1
The T(6,2) = 11 sets are: {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {2,3}, {2,5}, {3,4}, {3,5}, {4,5}, {5,6}. Missing from this list are: {2,4}, {2,6}, {3,6}, {4,6}.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A085945.
Second column is A015614.
Cf. A000837, A186974, A187106, A289508,  A289509, A300486, A303139, A320424, A320436.

Table[Length[Select[Subsets[Range[n],{k}],GCD@@#==1&]],{n,10},{k,n}]

T(n,k) = sum(d=1, n\k, moebius(d)*binomial(n\d, k)) \\ Andrew Howroyd, Jan 19 2023

T(n,k) = Sum_{d=1..floor(n/k)} mu(d)*binomial(floor(n/d), k). - Andrew Howroyd, Jan 19 2023

A302979 Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 15, 16, 22, 26, 30, 32, 33, 34, 35, 36, 38, 42, 46, 51, 55, 58, 62, 64, 66, 69, 70, 74, 77, 78, 82, 85, 86, 93, 94, 95, 100, 102, 105, 106, 110, 114, 118, 119, 122, 123, 128, 130, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165

Offset: 1

Gus Wiseman, Apr 16 2018

nonn

A prime index of n is a number m such that prime(m) divides n. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The number of uniform partitions of n with relatively prime parts is A078374(n).

			Sequence of all uniform relatively prime integer partitions begins (1), (11), (21), (111), (31), (41), (32), (1111), (51), (61), (321), (11111), (52), (71), (43), (2211).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796.

Select[Range[200],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]===1,SameQ@@FactorInteger[#][[All,2]]]&]

cp's OEIS Frontend

A302094 Number of relatively prime or monic twice-partitions of n.

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A302915 Number of relatively prime enriched p-trees of weight n.

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A302916 Number of relatively prime p-trees of weight n.

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A302917 Solution to a(1) = 1 and Sum_y Product_i a(y_i) = 0 for each n > 1, where the sum is over all relatively prime or monic partitions of n.

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A320435 Regular triangle read by rows where T(n,k) is the number of relatively prime k-subsets of {1,...,n}, 1 <= k <= n.

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A302979 Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.

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