A367584 Least 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

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..59.

Formula

a(p) = p for all primes p.

Example

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}

Mathematica

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]}]

Crossrefs

Positions of primes are A000040.

Positions of squarefree numbers are A000961.

All terms are rootless A007916.

Contains no nonprime prime powers A246547.

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

Positions of first appearances in A367580.

The sorted version is A367585.

The complement is A367768.

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. A005117, A020639, A051904, A072774, A130091, A367586.

Sequence in context: A172989 A095113 A345179 * A002517 A253568 A053570

Adjacent sequences: A367581 A367582 A367583 * A367585 A367586 A367587

Keyword

nonn

Author

Gus Wiseman, Nov 29 2023

Status

approved