A362982 - OEIS

A362982

Heinz numbers of partitions such that 2*(least part) < greatest part.

10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94, 95, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 124, 126

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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.

EXAMPLE

The terms together with their prime indices begin:

10: {1,3} 44: {1,1,5} 70: {1,3,4}

14: {1,4} 46: {1,9} 74: {1,12}

20: {1,1,3} 50: {1,3,3} 76: {1,1,8}

22: {1,5} 51: {2,7} 78: {1,2,6}

26: {1,6} 52: {1,1,6} 80: {1,1,1,1,3}

28: {1,1,4} 56: {1,1,1,4} 82: {1,13}

30: {1,2,3} 57: {2,8} 84: {1,1,2,4}

33: {2,5} 58: {1,10} 85: {3,7}

34: {1,7} 60: {1,1,2,3} 86: {1,14}

38: {1,8} 62: {1,11} 87: {2,10}

39: {2,6} 66: {1,2,5} 88: {1,1,1,5}

40: {1,1,1,3} 68: {1,1,7} 90: {1,2,2,3}

42: {1,2,4} 69: {2,9} 92: {1,1,9}

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100], 2*Min@@prix[#]<Max@@prix[#]&]

CROSSREFS

For prime factors instead of indices we have A069900, complement A081306.

Prime indices are listed by A112798, length A001222, sum A056239.

Partitions of this type are counted by A237820.

The complement is A362981, counted by A237824.

Cf. A027746, A053263, A171979, A237821, A327473, A327476, A362616, A362619, A362621, A362622.

Sequence in context: A073493 A162685 A272374 * A251727 A253785 A245729

Adjacent sequences: A362979 A362980 A362981 * A362983 A362984 A362985

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 14 2023

STATUS

approved