A354235 Heinz numbers of integer partitions with at least one part divisible by 3.

5, 10, 13, 15, 20, 23, 25, 26, 30, 35, 37, 39, 40, 45, 46, 47, 50, 52, 55, 60, 61, 65, 69, 70, 73, 74, 75, 78, 80, 85, 89, 90, 91, 92, 94, 95, 100, 103, 104, 105, 110, 111, 113, 115, 117, 120, 122, 125, 130, 135, 137, 138, 140, 141, 143, 145, 146, 148, 150

Offset: 1

Author

Gus Wiseman, May 23 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:
    5: {3}
   10: {1,3}
   13: {6}
   15: {2,3}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   26: {1,6}
   30: {1,2,3}
   35: {3,4}
   37: {12}
   39: {2,6}
   40: {1,1,1,3}
   45: {2,2,3}
   46: {1,9}
   47: {15}
   50: {1,3,3}
   52: {1,1,6}
   55: {3,5}
   60: {1,1,2,3}
		

Crossrefs

For 4 instead of 3 we have A046101, counted by A295342.

This sequence ranks the partitions counted by A295341, compositions A335464.

For 2 instead of 3 we have A324929 (and A013929), counted by A047967.

A001222 counts prime factors with multiplicity, distinct A001221.

A004250 counts partitions with some part > 2, compositions A008466.

A004709 lists numbers divisible by no cube, counted by A000726.

A036966 lists 3-full numbers, counted by A100405.

A046099 lists non-cubefree numbers.

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

A124010 gives prime signature, sorted A118914.

A354234 counts partitions of n with at least one part divisible by k.

Cf. A000720, A003963, A008483, A018256, A062739, A064428, A117485, A181819, A339222, A353508.

Programs

Mathematica

Select[Range[100],MemberQ[PrimePi/@First/@If[#==1,{},FactorInteger[#]]/3,_?IntegerQ]&]