A333219 Heinz number of the n-th composition in standard order.

1, 2, 3, 4, 5, 6, 6, 8, 7, 10, 9, 12, 10, 12, 12, 16, 11, 14, 15, 20, 15, 18, 18, 24, 14, 20, 18, 24, 20, 24, 24, 32, 13, 22, 21, 28, 25, 30, 30, 40, 21, 30, 27, 36, 30, 36, 36, 48, 22, 28, 30, 40, 30, 36, 36, 48, 28, 40, 36, 48, 40, 48, 48, 64, 17, 26, 33, 44

Offset: 1

Author

Gus Wiseman, Mar 16 2020

Keywords

• nonn

Comments

Includes all positive integers.

The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

The Heinz number of a composition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
   1: {}           15: {2,3}          25: {3,3}
   2: {1}          20: {1,1,3}        30: {1,2,3}
   3: {2}          15: {2,3}          30: {1,2,3}
   4: {1,1}        18: {1,2,2}        40: {1,1,1,3}
   5: {3}          18: {1,2,2}        21: {2,4}
   6: {1,2}        24: {1,1,1,2}      30: {1,2,3}
   6: {1,2}        14: {1,4}          27: {2,2,2}
   8: {1,1,1}      20: {1,1,3}        36: {1,1,2,2}
   7: {4}          18: {1,2,2}        30: {1,2,3}
  10: {1,3}        24: {1,1,1,2}      36: {1,1,2,2}
   9: {2,2}        20: {1,1,3}        36: {1,1,2,2}
  12: {1,1,2}      24: {1,1,1,2}      48: {1,1,1,1,2}
  10: {1,3}        24: {1,1,1,2}      22: {1,5}
  12: {1,1,2}      32: {1,1,1,1,1}    28: {1,1,4}
  12: {1,1,2}      13: {6}            30: {1,2,3}
  16: {1,1,1,1}    22: {1,5}          40: {1,1,1,3}
  11: {5}          21: {2,4}          30: {1,2,3}
  14: {1,4}        28: {1,1,4}        36: {1,1,2,2}
		

Crossrefs

The length of the k-th composition in standard order is A000120(k).

The sum of the k-th composition in standard order is A070939(k).

The maximum of the k-th composition in standard order is A070939(k).

A partial inverse is A333220. See also A233249.

Cf. A048793, A056239, A066099, A112798, A114994, A124767, A213925, A225620, A228351, A233564, A272919, A333218.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Times@@Prime/@stc[n],{n,0,100}]

Formula

A056239(a(n)) = A070939(n).