A302505 -id:A302505 - cp's OEIS frontend

A302569 Numbers that are either prime or whose prime indices are pairwise coprime. Heinz numbers of integer partitions with pairwise coprime parts.

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

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

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
08: {{},{},{}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
13: {{1,2}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
19: {{1,1,1}}
20: {{},{},{2}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}
26: {{},{1,2}}
28: {{},{},{1,1}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
32: {{},{},{},{},{}}

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A122132.

Cf. A000961, A001222, A005117, A007359, A007716, A051424, A056239, A076610, A101268, A275024, A302505, A302568.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]

is(n)=if(n<9, return(n>1)); n>>=valuation(n,2); if(n<9, return(1)); my(f=factor(n)); if(vecmax(f[,2])>1, return(0)); if(#f~==1, return(1)); my(v=apply(primepi, f[,1]),P=vecprod(v)); for(i=1,#v, if(gcd(v[i],P/v[i])>1, return(0))); 1 \\ Charles R Greathouse IV, Nov 11 2021

A303975 Number of distinct prime factors in the product of prime indices of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 0, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 28 2018

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.

First appearance of n is A062447(n - 1).

			9193 has prime indices {10, 66} with product 660, which has 4 distinct prime factors {2, 3, 5, 11}, so a(9193) = 4.

Cf. A000961, A001055, A001221, A002110, A003963, A056239, A112798, A156061, A302242, A302505, A320325.

Table[PrimeNu[If[n==1,1,Times@@PrimePi/@First/@FactorInteger[n]]],{n,100}]

a(n) = my(v = factor(n)[, 1]); omega(prod(i = 1, #v, primepi(v[i]))) \\ David A. Corneth, Dec 29 2018

a(n) = A001221(A156061(n)). - Michel Marcus, Jan 01 2019

A302796 Squarefree numbers whose prime indices are relatively prime. Nonprime Heinz numbers of strict integer partitions with relatively prime parts.

Original entry on oeis.org

1, 2, 6, 10, 14, 15, 22, 26, 30, 33, 34, 35, 38, 42, 46, 51, 55, 58, 62, 66, 69, 70, 74, 77, 78, 82, 85, 86, 93, 94, 95, 102, 105, 106, 110, 114, 118, 119, 122, 123, 130, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165, 166, 170, 174, 177, 178, 182

Offset: 1

Gus Wiseman, Apr 13 2018

nonn

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

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

			Sequence of terms together with their sets of prime indices begins:
01 : {}
02 : {1}
06 : {1,2}
10 : {1,3}
14 : {1,4}
15 : {2,3}
22 : {1,5}
26 : {1,6}
30 : {1,2,3}
33 : {2,5}
34 : {1,7}
35 : {3,4}
38 : {1,8}
42 : {1,2,4}
46 : {1,9}
51 : {2,7}
55 : {3,5}
58 : {1,10}
62 : {1,11}
66 : {1,2,5}

Cf. A001222, A003963, A005117, A007359, A051424, A056239, A275024, A289509, A302242, A302505, A302696, A302697, A302698, A302797, A302798.

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

isok(n) = {if (n == 1, return (1)); if (issquarefree(n), my(f = factor(n)); return (gcd(vector(#f~, k, primepi(f[k,1]))) == 1););} \\ Michel Marcus, Apr 13 2018

A302797 Squarefree numbers whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions with pairwise coprime parts.

Original entry on oeis.org

1, 2, 6, 10, 14, 15, 22, 26, 30, 33, 34, 35, 38, 46, 51, 55, 58, 62, 66, 69, 70, 74, 77, 82, 85, 86, 93, 94, 95, 102, 106, 110, 118, 119, 122, 123, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 161, 165, 166, 170, 177, 178, 186, 187, 190, 194, 201, 202

Offset: 1

Gus Wiseman, Apr 13 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. A single number is not considered coprime unless it is equal to 1.

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

			Sequence of terms together with their sets of prime indices begins:
01 : {}
02 : {1}
06 : {1,2}
10 : {1,3}
14 : {1,4}
15 : {2,3}
22 : {1,5}
26 : {1,6}
30 : {1,2,3}
33 : {2,5}
34 : {1,7}
35 : {3,4}
38 : {1,8}
46 : {1,9}
51 : {2,7}
55 : {3,5}
58 : {1,10}
62 : {1,11}
66 : {1,2,5}
69 : {2,9}
70 : {1,3,4}

Cf. A001222, A003963, A005117, A007359, A051424, A056239, A275024, A289509, A302242, A302505, A302696, A302697, A302698, A302796, A302798.

Select[Range[100],Or[#===1,SquareFreeQ[#]&&CoprimeQ@@PrimePi/@FactorInteger[#][[All,1]]]&]

A322526 Number of integer partitions of n whose product of parts is a squarefree number.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 6, 8, 9, 10, 13, 15, 17, 21, 24, 27, 30, 36, 41, 46, 51, 57, 65, 73, 82, 90, 101, 109, 121, 134, 150, 164, 177, 193, 214, 232, 253, 278, 300, 324, 351, 386, 419, 452, 484, 521, 563, 610, 658, 706, 758, 809, 868, 938, 1006, 1071, 1140, 1220, 1307

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

The parts of such a partition must also be squarefree and distinct except for any number of 1's.

			The a(8) = 9 partitions are (53), (71), (521), (611), (5111), (32111), (311111), (2111111), (11111111). Missing from this list are (8), (62), (44), (431), (422), (4211), (41111), (332), (3311), (3221), (2222), (22211), (221111).

Partial sums of A322530.

Cf. A001055, A003963, A005117, A064573 , A073576, A096276, A301987, A302505, A319005, A319916, A320322, A322527, A322529, A322531.

Table[Length[Select[IntegerPartitions[n],SquareFreeQ[Times@@#]&]],{n,30}]

A302521 Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 41, 43, 47, 51, 55, 59, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 215, 219, 221, 233, 241, 249, 255, 257, 269, 271

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

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

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set partitions.
01: {}
03: {{1}}
05: {{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
55: {{2},{3}}
59: {{7}}
67: {{8}}
73: {{2,4}}
79: {{1,5}}
83: {{9}}
85: {{2},{4}}
93: {{1},{5}}

Cf. A000110, A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243, A302505.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],UnsameQ@@Join@@primeMS/@primeMS[#]&]

A302798 Squarefree numbers that are prime or whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions that either consist of a single part or have pairwise coprime parts.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 66, 67, 69, 70, 71, 73, 74, 77, 79, 82, 83, 85, 86, 89, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110, 113, 118, 119, 122

Offset: 1

Gus Wiseman, Apr 13 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. A single number is not considered coprime unless it is equal to 1.

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

			Sequence of terms together with their sets of prime indices begins:
01 : {}
02 : {1}
03 : {2}
05 : {3}
06 : {1,2}
07 : {4}
10 : {1,3}
11 : {5}
13 : {6}
14 : {1,4}
15 : {2,3}
17 : {7}
19 : {8}
22 : {1,5}
23 : {9}
26 : {1,6}
29 : {10}
30 : {1,2,3}

Cf. A001222, A003963, A005117, A007359, A051424, A056239, A275024, A289509, A294472, A302242, A302505, A302696, A302697, A302698, A302796, A302797.

Select[Range[100],Or[#===1,SquareFreeQ[#]&&(PrimeQ[#]||CoprimeQ@@PrimePi/@FactorInteger[#][[All,1]])]&]

A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 31, 34, 45, 51, 63, 72, 88, 97, 120, 128, 158, 174, 201, 222, 264, 287, 333, 359, 416, 441, 518, 557, 631, 684, 770, 833, 954, 1017, 1141, 1222, 1378, 1475, 1643, 1755, 1939, 2097, 2327, 2471, 2758, 2928, 3233, 3470, 3813, 4085

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 18 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (52)       (44)
             (111)  (31)    (41)     (42)      (61)       (53)
                    (211)   (221)    (51)      (331)      (71)
                    (1111)  (311)    (222)     (421)      (422)
                            (2111)   (321)     (511)      (521)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (3211)     (2222)
                                     (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
Missing from the list for n = 7 through 9:
  (43)   (62)    (54)
  (322)  (332)   (63)
         (431)   (432)
         (3221)  (522)
                 (621)
                 (3222)
                 (3321)
                 (4311)
                 (32211)

Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.

Table[Length[Select[IntegerPartitions[n],SameQ@@Last/@FactorInteger[Times@@#]&]],{n,30}]

A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 3, 6, 5, 5, 5, 6, 8, 8, 9, 8, 11, 8, 12, 13, 16, 14, 13, 16, 21, 18, 21, 25, 22, 24, 27, 35, 33, 33, 32, 37, 42, 47, 48, 48, 52, 51, 59, 70, 68, 65, 69, 80, 87, 90, 103, 100, 96, 103, 123, 128, 135, 136, 132, 153

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

Such a partition must be strict and its parts must also be squarefree.

			The a(26) = 11 integer partitions:
  (26),
  (15,11), (19,7), (21,5), (23,3),
  (13,7,6), (13,10,3), (13,11,2), (17,7,2), (19,5,2),
  (11,7,5,3).

First differences of A322526.

Cf. A002865, A003963, A005117, A064573, A072774, A073576, A302505, A319005, A319057, A319169, A320322, A322527, A322528, A322529.

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

A324324 MM-numbers of crossing set partitions.

Original entry on oeis.org

2117, 3973, 4843, 5891, 6757, 7181, 7801, 10019, 10063, 11051, 11567, 13021, 13193, 13459, 14123, 14921, 17603, 18407, 18761, 18877, 19307, 19633, 20941, 21083, 21251, 21457, 22849, 23519, 23533, 24727, 26101, 27133, 27169, 27173, 27413, 29111, 30479, 31261

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part in the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is crossing if it contains two parts of the form {{...x...y...},{...z...t...}} with x < z < y < t or z < x < t < y.

Cf. A000108 (non-crossing set partitions), A001055, A001222, A003963, A005117, A016098 (crossing set partitions), A054726, A056239, A112798, A302242, A302243, A302505, A302521 (MM-numbers of set partitions).

Cf. A324166, A324167, A324169, A324170, A324171, A324326.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
setptnQ[bks_]:=UnsameQ@@Join@@bks&&!MemberQ[bks,{}];
Select[Range[10000],And[croXQ[primeMS/@primeMS[#]],setptnQ[primeMS/@primeMS[#]]]&]

cp's OEIS Frontend

A302569 Numbers that are either prime or whose prime indices are pairwise coprime. Heinz numbers of integer partitions with pairwise coprime parts.

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A303975 Number of distinct prime factors in the product of prime indices of n.

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A302796 Squarefree numbers whose prime indices are relatively prime. Nonprime Heinz numbers of strict integer partitions with relatively prime parts.

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A302797 Squarefree numbers whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions with pairwise coprime parts.

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A322526 Number of integer partitions of n whose product of parts is a squarefree number.

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A302521 Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.

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A302798 Squarefree numbers that are prime or whose prime indices are pairwise coprime. Heinz numbers of strict integer partitions that either consist of a single part or have pairwise coprime parts.

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A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

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A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

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