A353699 Heinz numbers of integer partitions whose product equals their length.

2, 6, 20, 36, 56, 176, 240, 416, 864, 1088, 1344, 2432, 3200, 5888, 8448, 14848, 23040, 31744, 35840, 39936, 75776, 167936, 208896, 331776, 352256, 450560, 516096, 770048, 802816, 933888, 1736704, 2457600, 3866624, 4259840, 4521984, 7995392, 12976128, 17563648

Offset: 1

Author

Gus Wiseman, May 19 2022

Keywords

• nonn

Comments

The Heinz number of a 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 terms together with their prime indices begin:
      2: {1}
      6: {1,2}
     20: {1,1,3}
     36: {1,1,2,2}
     56: {1,1,1,4}
    176: {1,1,1,1,5}
    240: {1,1,1,1,2,3}
    416: {1,1,1,1,1,6}
    864: {1,1,1,1,1,2,2,2}
   1088: {1,1,1,1,1,1,7}
   1344: {1,1,1,1,1,1,2,4}
   2432: {1,1,1,1,1,1,1,8}
   3200: {1,1,1,1,1,1,1,3,3}
   5888: {1,1,1,1,1,1,1,1,9}
   8448: {1,1,1,1,1,1,1,1,2,5}
  14848: {1,1,1,1,1,1,1,1,1,10}
  23040: {1,1,1,1,1,1,1,1,1,2,2,3}
  31744: {1,1,1,1,1,1,1,1,1,1,11}
  35840: {1,1,1,1,1,1,1,1,1,1,3,4}
  39936: {1,1,1,1,1,1,1,1,1,1,2,6}
  75776: {1,1,1,1,1,1,1,1,1,1,1,12}
		

Crossrefs

Length is A001222, counted by A008284, distinct A001221.

Product is A003963, counted by A339095, firsts A318871.

A similar sequence is A353503, counted by A353506.

These partitions are counted by A353698.

A005361 gives product of signature, firsts A353500 (sorted A085629).

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914.

A181819 gives prime shadow, with an inverse A181821.

A353394 gives product of shadows of prime indices, firsts A353397.

Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.

Programs

Mathematica

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]