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approved

proposed

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editing

proposed

Contains no non-prime nonprime prime powers A246547.

approved

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proposed

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Cf. A005117, A020639, `A051128, A051904, A072774, `A130091, ~A181819, A367586.

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.

The least number with multiset multiplicity kernel 6 is 12, so a(6) = 12.

Positions of primes (which are the only prime powers) are A000040.

The MMK triangle is A367579, sum A367581, min A055396, max A367583, min A367587.

Cf. `A001597, A005117, A020639, `A051128, A051904, `A052409, `A052410, A062770, A072774, `A130091, `A175781, ~A181819, ~A238747, ~A353742, A367582, A367685, `A367586.

The complement is A367768.

allocated for Gus WisemanLeast number whose multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) is n. First position of n in A367580.

1, 2, 3, 6, 5, 12, 7, 30, 15, 20, 11, 90, 13, 28, 45, 210, 17, 60, 19, 150, 63, 44, 23, 630, 35, 52, 105, 252, 29, 360, 31, 2310, 99, 68, 175, 2100, 37, 76, 117, 1050, 41, 504, 43, 396, 525, 92, 47, 6930, 77, 140, 153, 468, 53, 420, 275, 1470, 171, 116, 59

1,2

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.

We define the multiset multiplicity kernel (MMK) of a positive integer n to be the product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n. For example, 90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so MMK(90) = 12. As an operation on multisets, MMK is represented by the triangle A367579, and as an operation on their ranks it is represented by A367580.

a(p) = p for all primes p.

The least number with multiset multiplicity kernel 6 is 12, so a(6) = 12.

The least number with multiset multiplicity kernel 9 is 15, so a(9) = 15.

The terms together with their prime indices begin:

1 -> 1: {}

2 -> 2: {1}

3 -> 3: {2}

4 -> 6: {1,2}

5 -> 5: {3}

6 -> 12: {1,1,2}

7 -> 7: {4}

8 -> 30: {1,2,3}

9 -> 15: {2,3}

10 -> 20: {1,1,3}

11 -> 11: {5}

12 -> 90: {1,2,2,3}

13 -> 13: {6}

14 -> 28: {1,1,4}

15 -> 45: {2,2,3}

16 ->210: {1,2,3,4}

nn=1000;

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];

spnm[y_]:=Max@@NestWhile[Most, Sort[y], Union[#]!=Range[Max@@#]&];

qq=Table[Times@@mmk[Join@@ConstantArray@@@FactorInteger[n]], {n, nn}];

Table[Position[qq, i][[1, 1]], {i, spnm[qq]}]

Positions of primes (which are the only prime powers) are A000040.

Positions of squarefree numbers are A000961.

All terms are rootless A007916.

Contains no non-prime prime powers A246547.

The MMK triangle is A367579, sum A367581, max A367583, min A367587.

Positions of first appearances in A367580.

The sorted version is A367585.

A007947 gives squarefree kernel.

A027746 lists prime factors, length A001222, indices A112798.

A027748 lists distinct prime factors, length A001221, indices A304038.

A071625 counts distinct prime exponents.

A124010 gives prime signature, sorted A118914.

Cf. `A001597, A005117, A020639, `A051128, A051904, `A052409, `A052410, A062770, A072774, A130091, `A175781, A181819, ~A238747, ~A353742, A367582, A367685, `A367586.

allocated

nonn

Gus Wiseman, Nov 29 2023

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