A298748 -id:A298748 - cp's OEIS frontend

A305732 Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Gus Wiseman, Jun 22 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n. A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1,...,m_k) = 1 and the multiset {y_1,...,y_k} is also reducible.

			60 has relatively prime prime indices {1,1,2,3} with multiplicities {1,1,2} corresponding to A181819(90) = 12. 12 has relatively prime prime indices {1,1,2} with multiplicities {1,2} corresponding to A181819(12) = 6. 6 has relatively prime prime indices {1,2} with multiplicities {1,1} corresponding to A181819(6) = 4. 4 has relatively prime prime indices {1,1} with multiplicities {2} corresponding to A181819(4) = 3. 3 is prime, so we conclude that 60 belongs to the sequence.

Cf. A000837, A007916, A056239, A181819, A182850, A289508, A289509, A298748, A304465, A304687, A304818, A305563, A305731, A305733.

rdzQ[n_]:=And[n>1,Or[PrimeQ[n],And[rdzQ[Times@@Prime/@FactorInteger[n][[All,2]]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1]]];
Select[Range[50],rdzQ]

A303139 Number of integer partitions of n with at least two but not all parts having a common divisor greater than 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 6, 13, 17, 33, 37, 68, 82, 125, 159, 237, 278, 409, 491, 674, 830, 1121, 1329, 1781, 2144, 2770, 3345, 4299, 5086, 6507, 7752, 9687, 11571, 14378, 16985, 21039, 24876, 30379, 35924, 43734, 51320, 62238, 73068, 87747, 103021, 123347, 143955

Offset: 1

Gus Wiseman, Apr 19 2018

nonn

			The a(7) = 5 partitions are (421), (331), (322), (2221), (22111).

Cf. A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302569, A302696, A302796, A303138, A303140.

Table[Select[IntegerPartitions[n],!CoprimeQ@@#&&GCD@@#===1&]//Length,{n,30}]

A303280 Number of strict integer partitions of n whose parts have a common divisor other than 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 1, 10, 1, 11, 6, 12, 1, 19, 3, 18, 8, 23, 1, 36, 1, 32, 13, 38, 7, 57, 1, 54, 19, 68, 1, 95, 1, 90, 33, 104, 1, 148, 5, 149, 39, 166, 1, 230, 14, 226, 55, 256, 1, 360, 1, 340, 82, 390, 20, 527, 1, 513, 105, 609, 1

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

			The a(18) = 10 strict partitions are (18), (10,8), (12,6), (14,4), (15,3), (16,2), (8,6,4), (9,6,3), (10,6,2), (12,4,2).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000009, A000837, A018783, A051424, A078374, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303138.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> -add(mobius(d)*b(n/d), d=divisors(n) minus {1}):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 23 2018

Table[-Sum[MoebiusMu[d]*PartitionsQ[n/d],{d,Rest[Divisors[n]]}],{n,100}]

a(n) = -Sum_{d|n, d > 1} mu(d) * A000009(n/d).

A319161 Numbers whose prime multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

Numbers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (), (1), (2), (11), (3), (4), (111), (22), (5), (211), (6), (1111), (7), (221), (8), (311), (9), (2111), (33), (222), (411).

Cf. A001597, A007916, A056239, A072774, A181819, A182850, A289509, A296150, A298748, A305732, A319151, A319160, A319163, A319165.

Select[Range[100],GCD@@Length/@Split[Sort[FactorInteger[#][[All,2]]]]==1&]

A303283 Squarefree numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

Original entry on oeis.org

42, 78, 105, 114, 130, 174, 182, 195, 210, 222, 230, 231, 258, 266, 285, 318, 345, 357, 366, 370, 390, 406, 426, 429, 435, 455, 462, 470, 474, 483, 494, 518, 534, 546, 555, 570, 598, 602, 606, 610, 627, 638, 642, 645, 651, 663, 665, 678, 690, 705, 714, 715

Offset: 1

Gus Wiseman, Apr 20 2018

nonn

A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.

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

			The sequence of strict integer partitions whose Heinz numbers belong to this sequence begins (4,2,1), (6,2,1), (4,3,2), (8,2,1), (6,3,1), (10,2,1), (6,4,1), (6,3,2), (4,3,2,1), (12,2,1), (9,3,1), (5,4,2), (14,2,1), (8,4,1), (8,3,2), (16,2,1), (9,3,2), (7,4,2), (18,2,1), (12,3,1), (6,3,2,1).

Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303282.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],SquareFreeQ[#]&&!CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]

A319151 Heinz numbers of superperiodic integer partitions.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

First differs from A061345 at a(1) = 2 and next at a(98) = 441.
A number n is in the sequence iff n = 2 or the prime indices of n have a common divisor > 1 and the Heinz number of the multiset of prime multiplicities of n, namely A181819(n), is already in the sequence.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of partitions whose Heinz numbers belong to the sequence begins: (1), (2), (3), (4), (2,2), (5), (6), (7), (8), (9), (3,3), (2,2,2), (10), (11), (12), (13), (14), (15), (4,4), (16), (17), (18), (19), (20), (21), (22), (2,2,2,2).

Vincenzo Librandi, Table of n, a(n) for n = 1..2326

Cf. A001597, A056239, A072774, A098859, A181819, A182850, A289509, A296150, A298748, A304464, A305732, A317246, A317257, A319149, A319152, A319157.

supperQ[n_]:=Or[n==2,And[GCD@@PrimePi/@FactorInteger[n][[All,1]]>1,supperQ[Times@@Prime/@FactorInteger[n][[All,2]]]]];
Select[Range[500],supperQ]

A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 0, 2, 0, 1

Offset: 1

Gus Wiseman, Apr 19 2018

			Triangle begins:
01:   1
02:   0  1
03:   1  0  1
04:   1  0  0  1
05:   2  0  0  0  1
06:   2  1  0  0  0  1
07:   4  0  0  0  0  0  1
08:   4  1  0  0  0  0  0  1
09:   6  0  1  0  0  0  0  0  1
10:   7  2  0  0  0  0  0  0  0  1
11:  11  0  0  0  0  0  0  0  0  0  1
12:  10  2  1  1  0  0  0  0  0  0  0  1
13:  17  0  0  0  0  0  0  0  0  0  0  0  1
14:  17  4  0  0  0  0  0  0  0  0  0  0  0  1
15:  23  0  2  0  1  0  0  0  0  0  0  0  0  0  1
The strict partitions counted in row 12 are the following.
T(12,1) = 10: (11,1) (9,2,1) (8,3,1) (7,5) (7,4,1) (7,3,2) (6,5,1) (6,3,2,1) (5,4,3) (5,4,2,1)
T(12,2) = 2:  (10,2) (6,4,2)
T(12,3) = 1:  (9,3)
T(12,4) = 1:  (8,4)
T(12,12) = 1: (12)

First column is A078374. Second column at even indices is same as first column. Row sums are A000009. Row sums with first column removed are A303280.

Cf. A000009, A000837, A018783, A051424, A117408, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303139, A303140, A303280.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#===k&]],{n,15},{k,n}]

If k divides n, T(n,k) = A078374(n/k); otherwise T(n,k) = 0.

A305733 Heinz numbers of irreducible integer partitions. Nonprime numbers whose prime indices have a common divisor > 1 or such that A181819(n) is already in the sequence.

Original entry on oeis.org

1, 9, 21, 25, 27, 36, 39, 49, 57, 63, 65, 81, 87, 91, 100, 111, 115, 117, 121, 125, 129, 133, 144, 147, 159, 169, 171, 183, 185, 189, 196, 203, 213, 216, 225, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 324, 325, 333, 339, 343

Offset: 1

Gus Wiseman, Jun 22 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n. A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1, ..., m_k) = 1 and the multiset {y_1, ..., y_k} is also reducible.

Cf. A000837, A007916, A056239, A181819, A182850, A289508, A289509, A298748, A305563, A305731, A305732.

rdzQ[n_]:=And[n>1,Or[PrimeQ[n],And[rdzQ[Times@@Prime/@FactorInteger[n][[All,2]]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1]]];
Select[Range[50],!rdzQ[#]&]

A319163 Perfect powers whose prime multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 144, 169, 243, 256, 289, 324, 343, 361, 400, 512, 529, 576, 625, 729, 784, 841, 961, 1024, 1331, 1369, 1600, 1681, 1728, 1849, 1936, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2704, 2809, 2916, 3125

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

Perfect powers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (22), (1111), (33), (222), (11111), (44), (111111), (2222), (55), (333), (1111111), (221111), (66), (22222), (11111111), (77), (222211), (444), (88), (331111), (111111111), (99), (22111111), (3333), (222222), (441111).

Cf. A001597, A007916, A056239, A072774, A181819, A289509, A296150, A298748, A319151, A319161, A319162, A319165.

Select[Range[1000],And[GCD@@FactorInteger[#][[All,2]]>1,GCD@@Length/@Split[Sort[FactorInteger[#][[All,2]]]]==1]&]

A319165 Perfect powers whose prime indices are not relatively prime.

Original entry on oeis.org

9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1331, 1369, 1521, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3249, 3481, 3721, 3969, 4225, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7569, 7921, 8281, 9261, 9409

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (22), (33), (222), (44), (2222), (55), (333), (66), (22222), (77), (444), (88), (4422), (99), (3333), (222222).

Cf. A001597, A056239, A072774, A181819, A289509, A296150, A298748, A319151, A319161, A319163, A319164.

Select[Range[10000],With[{t=Transpose[FactorInteger[#]]},And[GCD@@PrimePi/@t[[1]]>1,GCD@@t[[2]]>1]]&]

cp's OEIS Frontend

A305732 Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence.

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A303139 Number of integer partitions of n with at least two but not all parts having a common divisor greater than 1.

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A303280 Number of strict integer partitions of n whose parts have a common divisor other than 1.

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A319161 Numbers whose prime multiplicities appear with relatively prime multiplicities.

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A303283 Squarefree numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime.

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A319151 Heinz numbers of superperiodic integer partitions.

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A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

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A305733 Heinz numbers of irreducible integer partitions. Nonprime numbers whose prime indices have a common divisor > 1 or such that A181819(n) is already in the sequence.

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A319163 Perfect powers whose prime multiplicities appear with relatively prime multiplicities.

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