A344416 Heinz numbers of integer partitions whose sum is even and is at most twice the greatest part.

3, 4, 7, 9, 10, 12, 13, 19, 21, 22, 25, 28, 29, 30, 34, 37, 39, 40, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 76, 79, 82, 84, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146

Offset: 1

Author

Gus Wiseman, May 20 2021

Keywords

• nonn

Comments

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

Also numbers m whose sum of prime indices A056239(m) is even and is at most twice the greatest prime index A061395(m).

Examples

			The sequence of terms together with their prime indices begins:
      3: {2}         37: {12}          71: {20}
      4: {1,1}       39: {2,6}         76: {1,1,8}
      7: {4}         40: {1,1,1,3}     79: {22}
      9: {2,2}       43: {14}          82: {1,13}
     10: {1,3}       46: {1,9}         84: {1,1,2,4}
     12: {1,1,2}     49: {4,4}         85: {3,7}
     13: {6}         52: {1,1,6}       87: {2,10}
     19: {8}         53: {16}          88: {1,1,1,5}
     21: {2,4}       55: {3,5}         89: {24}
     22: {1,5}       57: {2,8}         91: {4,6}
     25: {3,3}       61: {18}          94: {1,15}
     28: {1,1,4}     62: {1,11}       101: {26}
     29: {10}        63: {2,2,4}      102: {1,2,7}
     30: {1,2,3}     66: {1,2,5}      107: {28}
     34: {1,7}       70: {1,3,4}      111: {2,12}
		

Crossrefs

These partitions are counted by A000070 = even-indexed terms of A025065.

The opposite version appears to be A320924, counted by A209816.

The opposite version with odd weights allowed appears to be A322109.

The conjugate opposite version allowing odds is A344291, counted by A110618.

The conjugate version is A344296, also counted by A025065.

The conjugate opposite version is A344413, counted by A209816.

Allowing odd weight gives A344414.

The case of equality is A344415, counted by A035363.

A001222 counts prime factors with multiplicity.

A027187 counts partitions of even length, ranked by A028260.

A056239 adds up prime indices, row sums of A112798.

A058696 counts partitions of even numbers, ranked by A300061.

A265640 lists Heinz numbers of palindromic partitions.

A301987 lists numbers whose sum of prime indices equals their product.

A334201 adds up all prime indices except the greatest.

A340387 lists Heinz numbers of partitions whose sum is twice their length.

Cf. A001414, A074761, A316413, A316428, A325037, A325038, A325044, A330950, A344293, A344294, A344297.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[Total[primeMS[#]]]&&Max[primeMS[#]]>=Total[primeMS[#]]/2&]

Formula

Intersection of A300061 and A344414.