A337694 -id:A337694 - cp's OEIS frontend

A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376

Offset: 0

Gus Wiseman, Oct 24 2020

nonn

			The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
  2   3   4    5    6     7     8      9      A       B       C
          22   32   33    43    44     54     55      65      66
                    222   52    53     72     73      74      75
                          322   332    333    433     83      444
                                2222   522    532     92      543
                                       3222   3322    443     552
                                              22222   533     732
                                                      722     3333
                                                      3332    5322
                                                      5222    33222
                                                      32222   222222

A007359 (A302568) gives the strict case.
A101268 (A335235) gives pairwise coprime or singleton compositions.
A200976 (A338318) gives the pairwise non-coprime instead of coprime version.
A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).
A304712 (A338331) allows 1's, with strict version A007360 (A302798).
A327516 (A302696) gives pairwise coprime partitions.
A328673 (A328867) gives partitions with no distinct relatively prime parts.
A338315 (A337987) does not consider singletons coprime.
A338317 (A338316) gives these partitions.
A337462 (A333227) gives pairwise coprime compositions.
A337485 (A337984) gives pairwise coprime integer partitions with no 1's.
A337665 (A333228) gives compositions with pairwise coprime distinct parts.
A337667 (A337666) gives pairwise non-coprime compositions.
A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.
A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).
Cf. A051424, A289508, A302569, A303140, A337694.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&(SameQ@@#||CoprimeQ@@Union[#])&]],{n,0,15}]

The Heinz numbers of these partitions are given by A338316. 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.

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&]

A338318 Composite numbers whose prime indices are pairwise intersecting (non-coprime).

Original entry on oeis.org

9, 21, 25, 27, 39, 49, 57, 63, 65, 81, 87, 91, 111, 115, 117, 121, 125, 129, 133, 147, 159, 169, 171, 183, 185, 189, 203, 213, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 325, 333, 339, 343, 351, 361, 365, 371, 377, 387, 393

Offset: 1

Gus Wiseman, Oct 31 2020

nonn

First differs from A322336 in lacking 2535, with prime indices {2,3,6,6}.
First differs from A327685 in having 17719, with prime indices {6,10,15}.
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 pairwise intersecting (non-coprime) partitions with more than one part. 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 sequence of terms together with their prime indices begins:
      9: {2,2}        121: {5,5}        243: {2,2,2,2,2}
     21: {2,4}        125: {3,3,3}      247: {6,8}
     25: {3,3}        129: {2,14}       259: {4,12}
     27: {2,2,2}      133: {4,8}        261: {2,2,10}
     39: {2,6}        147: {2,4,4}      267: {2,24}
     49: {4,4}        159: {2,16}       273: {2,4,6}
     57: {2,8}        169: {6,6}        289: {7,7}
     63: {2,2,4}      171: {2,2,8}      299: {6,9}
     65: {3,6}        183: {2,18}       301: {4,14}
     81: {2,2,2,2}    185: {3,12}       303: {2,26}
     87: {2,10}       189: {2,2,2,4}    305: {3,18}
     91: {4,6}        203: {4,10}       319: {5,10}
    111: {2,12}       213: {2,20}       321: {2,28}
    115: {3,9}        235: {3,15}       325: {3,3,6}
    117: {2,2,6}      237: {2,22}       333: {2,2,12}

A200976 counts the partitions with these Heinz numbers.
A302696 is the pairwise coprime instead of pairwise non-coprime version.
A337694 includes the primes.
A002808 lists composite numbers.
A318717 counts pairwise intersecting strict partitions.
A328673 counts partitions with pairwise intersecting distinct parts, with Heinz numbers A328867 and restriction to triples A337599 (except n = 3).
Cf. A008578, A051185, A056239, A101268, A112798, A284825, A302569, A305843, A319752, A327516, A335236, A337666, A337667.

stabstrQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Select[Range[2,100],!PrimeQ[#]&&stabstrQ[PrimePi/@First/@FactorInteger[#],CoprimeQ]&]

Equals A337694 \ A008578.

cp's OEIS Frontend

A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

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

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A338318 Composite numbers whose prime indices are pairwise intersecting (non-coprime).

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