A344414 Heinz numbers of integer partitions whose sum is at most twice their greatest part.

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Author

Gus Wiseman, May 19 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.

Examples

			The sequence of terms together with their prime indices begins:
     2: {1}        20: {1,1,3}    39: {2,6}
     3: {2}        21: {2,4}      40: {1,1,1,3}
     4: {1,1}      22: {1,5}      41: {13}
     5: {3}        23: {9}        42: {1,2,4}
     6: {1,2}      25: {3,3}      43: {14}
     7: {4}        26: {1,6}      44: {1,1,5}
     9: {2,2}      28: {1,1,4}    46: {1,9}
    10: {1,3}      29: {10}       47: {15}
    11: {5}        30: {1,2,3}    49: {4,4}
    12: {1,1,2}    31: {11}       51: {2,7}
    13: {6}        33: {2,5}      52: {1,1,6}
    14: {1,4}      34: {1,7}      53: {16}
    15: {2,3}      35: {3,4}      55: {3,5}
    17: {7}        37: {12}       56: {1,1,1,4}
    19: {8}        38: {1,8}      57: {2,8}
For example, 56 has prime indices {1,1,1,4} and 7 <= 2*4, so 56 is in the sequence. On the other hand, 224 has prime indices {1,1,1,1,1,4} and 9 > 2*4, so 224 is not in the sequence.
		

Crossrefs

These partitions are counted by A025065 but are different from palindromic partitions, which have Heinz numbers A265640.

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

The opposite version appears to be A322109, counted by A110618.

The case of equality in the conjugate version is A340387.

The conjugate opposite version is A344291, counted by A110618.

The conjugate opposite 5-smooth case is A344293, counted by A266755.

The conjugate version is A344296, also counted by A025065.

The case of equality is A344415.

The even-weight case is A344416.

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.

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

A334201 adds up all prime indices except the greatest.

Cf. A001414, A067538, A301988, A316413, A316428, A325037, A325038, A325044, A330950, A344294, A344297.

Programs

Mathematica

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

Formula

A056239(a(n)) <= 2*A061395(a(n)).