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proposed

The version for divisors instead of factors is the complement of A355740.

The version for divisors instead of factors is A368110, complement A355740.

Cf. `A092918, A355737, A355739, A355741, A355744, A355745, `A367901, A367902, `A367904.

Positions of positive terms in A367771The complement is A355529, odd A355535, binary A367907.

Positions The version for divisors instead of factors is the complement of 0's in A367771 are A355529, odd A355535, binary A367907A355740.

Positions of 1's positive terms in A367771 are A368101, binary A367908.

For a unique choice we have A368101, binary A367908.

A124010 gives prime signature, sort sorted A118914, length A001221, sum A001222.

Cf. `A007716, A055621, `A083323, `A092918, `A300913, A355737, A355739, A355740, A355741, A355744, A355745, `A367901, A367902, `A367903, A367904, ~A367912.

Also MM-numbers A prime index of n is a number m such that prime(m) divides n. The multiset partitions satisfying a strict version of the axiom prime indices of n is row n of choiceA112798.

The prime indices of prime indices of 2849 are {4,5,12}, with prime factors {{1,12,2},{35},{1,1,2,2,3}}, and there are of the two choices (1,3,12,5,2) and (1,3,2,5,3), the latter has all different terms, so 2849 is in the sequence.

The prime indices of prime indices of 2849 are {{1,1},{3},{1,1,2}}, and there are two choices (1,3,1) and (1,3,2), so 2849 is in the sequence.

allocated for Gus WisemanNumbers of which it is possible to choose a different prime factor of each prime index.

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 127, 129, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163

1,2

Also MM-numbers of multiset partitions satisfying a strict version of the axiom of choice.

The terms together with their prime indices of prime indices begin:

1: {}

3: {{1}}

5: {{2}}

7: {{1,1}}

11: {{3}}

13: {{1,2}}

15: {{1},{2}}

17: {{4}}

19: {{1,1,1}}

23: {{2,2}}

29: {{1,3}}

31: {{5}}

33: {{1},{3}}

35: {{2},{1,1}}

37: {{1,1,2}}

39: {{1},{1,2}}

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

Select[Range[100], Select[Tuples[prix/@prix[#]], UnsameQ@@#&]!={}&]

Positions of positive terms in A367771

Positions of 0's in A367771 are A355529, odd A355535, binary A367907.

Positions of 1's in A367771 are A368101, binary A367908.

The version for binary indices is A367906, positive positions in A367905.

A058891 counts set-systems, covering A003465, connected A323818.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A124010 gives prime signature, sort A118914, length A001221, sum A001222.

Cf. `A007716, A055621, `A083323, `A092918, `A300913, A355737, A355739, A355740, A355741, A355744, A355745, `A367901, A367902, `A367903, A367904, ~A367912.

allocated

nonn

Gus Wiseman, Dec 12 2023

approved

editing

allocated for Gus Wiseman

allocated

approved