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proposed

For mode complement we have A356862, counted by A362608.

A362614 counts partitions by number of modes, ranked by A362611.

A362615 counts partitions by number of co-modes, ranked by A362613.

Cf. A000040, A000720, `A002865, `A098859, A215366, A327473, A327476, `A353864, `A353865, A356862, A359908, `A362612A362611.

allocated for Gus WisemanNumbers without a unique least prime exponent, or numbers whose prime factorization has more than one element of least multiplicity.

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

1,1

First differs from A130092 in lacking 180.

First differs from A351295 in lacking 180 and having 216.

First differs from A362605 in having 60.

The prime factorization of 1800 is {2,2,2,3,3,5,5}, and the parts of least multiplicity are {3,5}, so 1800 is in the sequence.

The terms together with their prime indices begin:

6: {1,2}

10: {1,3}

14: {1,4}

15: {2,3}

21: {2,4}

22: {1,5}

26: {1,6}

30: {1,2,3}

33: {2,5}

34: {1,7}

35: {3,4}

36: {1,1,2,2}

38: {1,8}

39: {2,6}

42: {1,2,4}

Select[Range[100], Count[Last/@FactorInteger[#], Min@@Last/@FactorInteger[#]]>1&]

For mode complement we have A356862, counted by A362608.

The complement is A359178, counted by A362610.

For mode we have A362605, counted by A362607.

Partitions of this type are counted by A362609.

These are the positions of terms > 1 in A362613.

A112798 lists prime indices, length A001222, sum A056239.

A362614 counts partitions by number of modes, ranked by A362611.

A362615 counts partitions by number of co-modes, ranked by A362613.

Cf. A000040, A000720, `A002865, `A098859, A215366, A327473, A327476, `A353864, `A353865, A359908, `A362612.

allocated

nonn

Gus Wiseman, May 05 2023

approved

editing

allocated for Gus Wiseman

allocated

approved