A337984 Heinz numbers of pairwise coprime integer partitions with no 1's, where a singleton is not considered coprime.

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 201, 205, 209, 215, 217, 219, 221, 249, 253, 255, 265, 287, 291, 295, 309, 323, 327, 329, 335, 341, 355, 381, 385, 391, 395, 403, 407, 411, 413, 415, 437, 447, 451, 465

Offset: 1

Author

Gus Wiseman, Oct 22 2020

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
     15: {2,3}     155: {3,11}     265: {3,16}
     33: {2,5}     161: {4,9}      287: {4,13}
     35: {3,4}     165: {2,3,5}    291: {2,25}
     51: {2,7}     177: {2,17}     295: {3,17}
     55: {3,5}     187: {5,7}      309: {2,27}
     69: {2,9}     201: {2,19}     323: {7,8}
     77: {4,5}     205: {3,13}     327: {2,29}
     85: {3,7}     209: {5,8}      329: {4,15}
     93: {2,11}    215: {3,14}     335: {3,19}
     95: {3,8}     217: {4,11}     341: {5,11}
    119: {4,7}     219: {2,21}     355: {3,20}
    123: {2,13}    221: {6,7}      381: {2,31}
    141: {2,15}    249: {2,23}     385: {3,4,5}
    143: {5,6}     253: {5,9}      391: {7,9}
    145: {3,10}    255: {2,3,7}    395: {3,22}
		

Crossrefs

A005117 is a superset.

A337485 counts these partitions.

A302568 considers singletons to be coprime.

A304711 allows 1's, with squarefree version A302797.

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

A007359 counts partitions into singleton or pairwise coprime parts with no 1's

A101268 counts pairwise coprime or singleton compositions, ranked by A335235.

A305713 counts pairwise coprime strict partitions.

A327516 counts pairwise coprime partitions, ranked by 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.

A337983 counts pairwise non-coprime strict compositions, with unordered version A318717 ranked by A318719.

Cf. A051424, A056239, A087087, A112798, A200976, A220377, A302569, A303140, A303282, A328673, A328867.

Programs

Mathematica

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

Formula

Equals A302568\A000040.