A335241 -id:A335241 - cp's OEIS frontend

A327516 Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.

1, 1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 22, 26, 32, 37, 42, 50, 59, 69, 80, 91, 101, 115, 133, 152, 170, 190, 210, 235, 265, 300, 334, 366, 398, 441, 484, 541, 597, 648, 703, 770, 848, 935, 1022, 1102, 1184, 1281, 1406, 1534, 1661, 1789, 1916, 2062, 2244, 2435

Offset: 0

Gus Wiseman, Sep 19 2019

nonn

The Heinz numbers of these partitions are given by A302696.

Note that the definition excludes partitions with repeated parts other than 1 (cf. A038348, A304709).

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (52)       (71)
                    (1111)  (311)    (411)     (61)       (431)
                            (2111)   (3111)    (511)      (521)
                            (11111)  (21111)   (3211)     (611)
                                     (111111)  (4111)     (5111)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750
Gus Wiseman, Sequences counting and encoding certain classes of multisets

Cf. A038348, A302569, A304709, A304711.
A000837 is the relatively prime instead of pairwise coprime version.
A051424 includes all singletons, with strict case A007360.
A101268 is the ordered version (with singletons).
A302696 ranks these partitions, with complement A335241.
A305713 is the strict case.
A307719 counts these partitions of length 3.
A018783 counts partitions with a common divisor.
A328673 counts pairwise non-coprime partitions.
Cf. A087087, A220377, A326675, A333227, A333228, A335238.

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

For n > 1, a(n) = A051424(n) - 1. - Gus Wiseman, Sep 18 2020

A336620 Numbers that are not a product of elements of A304711.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 42, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 78, 79, 81, 83, 87, 89, 91, 97, 101, 103, 105, 107, 109, 111, 113, 114, 115, 117, 121, 125, 126, 127, 129, 130, 131, 133, 137, 139, 147, 149

Offset: 1

Gus Wiseman, Aug 02 2020

nonn

A304711 lists numbers whose distinct prime indices are pairwise coprime.

The first term divisible by 4 is a(421) = 1092.

			The sequence of terms together with their prime indices begins:
      3: {2}         39: {2,6}       78: {1,2,6}
      5: {3}         41: {13}        79: {22}
      7: {4}         42: {1,2,4}     81: {2,2,2,2}
      9: {2,2}       43: {14}        83: {23}
     11: {5}         47: {15}        87: {2,10}
     13: {6}         49: {4,4}       89: {24}
     17: {7}         53: {16}        91: {4,6}
     19: {8}         57: {2,8}       97: {25}
     21: {2,4}       59: {17}       101: {26}
     23: {9}         61: {18}       103: {27}
     25: {3,3}       63: {2,2,4}    105: {2,3,4}
     27: {2,2,2}     65: {3,6}      107: {28}
     29: {10}        67: {19}       109: {29}
     31: {11}        71: {20}       111: {2,12}
     37: {12}        73: {21}       113: {30}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

A336426 is the version for superprimorials, with complement A181818.
A336497 is the version for superfactorials, with complement A336496.
A336735 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A007916, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A335241, A336424, A336568, A336736.

nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]=={}&]

cp's OEIS Frontend

A327516 Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.

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