A337987 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is not considered coprime unless it is (1).

15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 93, 95, 99, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 265, 275, 279, 287, 291, 295, 297, 309, 323, 327, 329, 335, 341, 355, 363, 369

Offset: 1

Author

Gus Wiseman, Oct 23 2020

Keywords

• nonn

Comments

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 integer partitions with no 1's whose distinct parts are pairwise coprime (A338315). 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.

Examples

			The sequence of terms together with their prime indices begins:
     15: {2,3}      135: {2,2,2,3}    215: {3,14}
     33: {2,5}      141: {2,15}       217: {4,11}
     35: {3,4}      143: {5,6}        219: {2,21}
     45: {2,2,3}    145: {3,10}       221: {6,7}
     51: {2,7}      153: {2,2,7}      225: {2,2,3,3}
     55: {3,5}      155: {3,11}       245: {3,4,4}
     69: {2,9}      161: {4,9}        249: {2,23}
     75: {2,3,3}    165: {2,3,5}      253: {5,9}
     77: {4,5}      175: {3,3,4}      255: {2,3,7}
     85: {3,7}      177: {2,17}       265: {3,16}
     93: {2,11}     187: {5,7}        275: {3,3,5}
     95: {3,8}      201: {2,19}       279: {2,2,11}
     99: {2,2,5}    205: {3,13}       287: {4,13}
    119: {4,7}      207: {2,2,9}      291: {2,25}
    123: {2,13}     209: {5,8}        295: {3,17}
		

Crossrefs

A304711 is the not necessarily odd version, with squarefree case A302797.

A337694 is a pairwise non-coprime instead of pairwise coprime version.

A337984 is the squarefree case.

A338315 counts the partitions with these Heinz numbers.

A338316 considers singletons coprime.

A007359 counts partitions into singleton or pairwise coprime parts with no 1's, with Heinz numbers A302568.

A304709 counts partitions whose distinct parts are pairwise coprime.

A327516 counts pairwise coprime partitions, with Heinz numbers A302696.

A337462 counts pairwise coprime compositions, ranked by A333227.

A337561 counts pairwise coprime strict compositions.

A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.

A337667 counts pairwise non-coprime compositions, ranked by A337666.

A337697 counts pairwise coprime compositions with no 1's.

A318717 counts pairwise non-coprime strict partitions, with Heinz numbers A318719.

Cf. A005408, A051424, A056239, A087087, A112798, A200976, A289508, A289509, A302569, A303282, A328867, A337485.

Programs

Mathematica

Select[Range[1,100,2],CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]