cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A367683 Numbers whose sorted prime signature is the same as the multiset multiplicity kernel of their prime indices.

Original entry on oeis.org

1, 2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 38, 40, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 102, 106, 110, 112, 114, 118, 122, 125, 126, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 225
Offset: 1

Views

Author

Gus Wiseman, Nov 30 2023

Keywords

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 A367579, and as an operation on their ranks it is represented by A367580.

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   34: {1,7}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   46: {1,9}
   58: {1,10}
   62: {1,11}
   66: {1,2,5}
   70: {1,3,4}

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..10000
  • Michael De Vlieger, 20 X 20 color coded list of terms, color function: gray = 1, red = prime, gold = composite prime power, green = squarefree composite, blue-purple = numbers neither squarefree nor prime powers. Bright green = primorial, light green = even squarefree semiprime, light blue = highly composite, middle blue = in A055932, purple = squareful but not a prime power.
  • Michael De Vlieger, 485 X 485 = 235225-term raster with the same color code as above.

Crossrefs

Squarefree terms are A039956.
The LHS is A118914, unsorted A124010.
Prime-power terms are A307539.
The RHS is A367579, ranks A367580, sum A367581, max A367583.
Partitions of this type are counted by A367682.
A007947 gives squarefree kernel.
A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, reversed A353742.
A304038 lists distinct prime indices, length A001221, sum A066328.
A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A000961, A005117, A051904, A052409, A071625, A072774, A130091, A367584, A367586, A367587.

Programs

  • Mathematica
    mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Select[Range[100], #==1||Sort[Last/@FactorInteger[#]] == mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]]&]

A095209 a(0) = 1, and for n > 0, a(n) = the least multiple of prime(n) such that the geometric mean of a(0) to a(n) is an integer.

Original entry on oeis.org

1, 4, 54, 3750, 504210, 372027810, 144949074270, 209481995953230, 164735296593157290, 401824316553919068810, 2721846739094340967339230, 5095936579799734140259818030, 48850362989361131638352534231610
Offset: 0

Views

Author

Amarnath Murthy, Jun 08 2004

Keywords

Examples

			(1*4*54*3750)^(1/4) = 30.

Links

Crossrefs

Cf. A000040, A002110, A020522, A057335, A095210, A307539.

Programs

Formula

From Antti Karttunen and Peter Munn, May 04 2022: (Start)
The n-th partial product of these terms = A002110(n)^(1+n), i.e., the n-th geometric mean is the n-th power of (n-1)-th primorial.
a(n) = A002110(n) * A307539(n).
a(n) = A057335(A020522(n)). [Found by LODA-miner, follows from the above formulas]
(End)

Extensions

Edited by Don Reble, Jan 06 2007
Starting offset changed from 1 to 0 and the definition accordingly edited by Antti Karttunen, May 04 2022

A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.

Original entry on oeis.org

1, 2, 4, 6, 9, 8, 12, 18, 20, 27, 30, 45, 50, 75, 125, 16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 2401, 32, 48, 72, 80, 108, 112, 120, 162
Offset: 0

Views

Author

Robert Price, Mar 03 2020

Keywords

Comments

Positive integers not in T are: 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, ... .
Row n has exactly one squarefree member: primorial(n) = A002110(n).
Sorting all terms (except 1) gives A324521.

Examples

			Triangle T(n,k) begins:
  1;
  2;
  4,  6,  9;
  8, 12, 18, 20, 27, 30, 45, 50, 75, 125;
  ...

Links

Crossrefs

Column k=1 gives A000079.
Last elements of rows give A307539.
Row lengths give A088218.
Row sums give A332967(n) = A124960(2n,n).
T(n,n) gives A101695(n) for n > 0.
Cf. A000720, A001222, A005117, A027748, A324521.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1], [seq(
      map(x-> x*ithprime(j), b(n-1, j))[], j=1..i)])
    end:
T:= n-> sort(b(n$2))[]:
seq(T(n), n=0..5);  # Alois P. Heinz, Mar 03 2020
  • Mathematica
    t = Table[Union[Apply[Times, Tuples[Prime[Range[n]], {n}], {1}]], {n, 0, 5}];
t // TableForm
Flatten[t]
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