A366318 Heinz numbers of integer partitions that are of length 2 or begin with n/2, where n is the sum of all parts.

4, 6, 9, 10, 12, 14, 15, 21, 22, 25, 26, 30, 33, 34, 35, 38, 39, 40, 46, 49, 51, 55, 57, 58, 62, 63, 65, 69, 70, 74, 77, 82, 84, 85, 86, 87, 91, 93, 94, 95, 106, 111, 112, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159

Offset

1,1

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.

Links

Table of n, a(n) for n=1..61.

Formula

Union of A001358 and A344415.

Example

The terms together with their prime indices begin:
     4: {1,1}      38: {1,8}         77: {4,5}
     6: {1,2}      39: {2,6}         82: {1,13}
     9: {2,2}      40: {1,1,1,3}     84: {1,1,2,4}
    10: {1,3}      46: {1,9}         85: {3,7}
    12: {1,1,2}    49: {4,4}         86: {1,14}
    14: {1,4}      51: {2,7}         87: {2,10}
    15: {2,3}      55: {3,5}         91: {4,6}
    21: {2,4}      57: {2,8}         93: {2,11}
    22: {1,5}      58: {1,10}        94: {1,15}
    25: {3,3}      62: {1,11}        95: {3,8}
    26: {1,6}      63: {2,2,4}      106: {1,16}
    30: {1,2,3}    65: {3,6}        111: {2,12}
    33: {2,5}      69: {2,9}        112: {1,1,1,1,4}
    34: {1,7}      70: {1,3,4}      115: {3,9}
    35: {3,4}      74: {1,12}       118: {1,17}

Mathematica

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

Crossrefs

The first condition alone is A001358, counted by A004526.

The complement of the first condition is A100959, counted by A058984.

The partitions with these Heinz numbers are counted by A238628.

The second condition alone is A344415, counted by A035363.

The complement of the second condition is A366319, counted by A086543.

A001222 counts prime factors with multiplicity.

A056239 adds up prime indices, row sums of A112798.

A322109 ranks partitions of n with no part > n/2, counted by A110618.

A334201 adds up all prime indices except the greatest.

A344296 solves for k in A001222(k) >= A056239(k)/2, counted by A025065.

Cf. A001414, A316413, A320924, A325044, A340387, A344414, A344416.

Sequence in context: A325270 A051278 A328028 * A339424 A175127 A174166

Adjacent sequences: A366315 A366316 A366317 * A366319 A366320 A366321

Keyword

nonn

Author

Gus Wiseman, Oct 08 2023

Status

approved