A319246 -id:A319246 - cp's OEIS frontend

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030

Offset: 0

Gus Wiseman, Dec 02 2020

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021
From David James Sycamore, Jan 09 2025: (Start)
Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.
Permutation of A005117, Squarefree version A379746. (End)

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Alois P. Heinz, Rows n = 0..14, flattened
Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

A011782 gives row lengths.
A339360 gives row sums.
A008578 (shifted) is column k = 1.
A100484 is column k = 2.
A001748 is column k = 3.
A002110 is column k = 2^(n-1).
A070826 is column k = 2^(n-1) - 1.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A261144 divides the n-th row by prime(n), with row sums A054640.
A339116 is the restriction to semiprimes, with row sums A339194.
A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
A006881 lists squarefree semiprimes.
A072047 counts prime factors of squarefree numbers.
A319246 is the sum of prime indices of the n-th squarefree number.
A329631 lists prime indices of squarefree numbers, reversed: A319247.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
Cf. A379746.

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42

Offset: 1

Jean-François Alcover, Nov 26 2015

If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021

			Triangle begins:
1, 2;                        squarefree and 2-smooth
1, 2, 3, 6;                  squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6,  7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...

Jean-François Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
Eric Weisstein's MathWorld, Smooth number.
Wikipedia, Smooth number

Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n-1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.

b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Nov 28 2015

primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021

A319247 Irregular triangle whose n-th row lists the strict integer partition whose Heinz number is the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 6, 4, 1, 3, 2, 7, 8, 4, 2, 5, 1, 9, 6, 1, 10, 3, 2, 1, 11, 5, 2, 7, 1, 4, 3, 12, 8, 1, 6, 2, 13, 4, 2, 1, 14, 9, 1, 15, 7, 2, 16, 5, 3, 8, 2, 10, 1, 17, 18, 11, 1, 6, 3, 5, 2, 1, 19, 9, 2, 4, 3, 1, 20, 21, 12, 1, 5, 4, 6, 2, 1, 22

Offset: 1

Gus Wiseman, Sep 15 2018

			The sequence of strict partitions begins: (), (1), (2), (3), (2,1), (4), (3,1), (5), (6), (4,1), (3,2), (7), (8), (4,2), (5,1), (9), (6,1), (10), (3,2,1), (11), (5,2), (7,1), (4,3), (12), (8,1), (6,2), (13), (4,2,1).

Row lengths are A072047. Rows sums are A319246.

Cf. A000009, A000720, A001222, A005117, A056239, A265668, A296150, A299755.

Table[If[n==1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,Select[Range[100],SquareFreeQ]}]

T(n,k) = A000720(A265668(n,-k)).

A329631 Irregular triangle read by rows where row n lists the prime indices of the n-th squarefree number.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 3, 5, 6, 1, 4, 2, 3, 7, 8, 2, 4, 1, 5, 9, 1, 6, 10, 1, 2, 3, 11, 2, 5, 1, 7, 3, 4, 12, 1, 8, 2, 6, 13, 1, 2, 4, 14, 1, 9, 15, 2, 7, 16, 3, 5, 2, 8, 1, 10, 17, 18, 1, 11, 3, 6, 1, 2, 5, 19, 2, 9, 1, 3, 4, 20, 21, 1, 12, 4, 5, 1, 2, 6

Offset: 1

Gus Wiseman, Nov 18 2019

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			Triangle begins:
   1: {}         33: {2,5}      66: {1,2,5}     97: {25}
   2: {1}        34: {1,7}      67: {19}       101: {26}
   3: {2}        35: {3,4}      69: {2,9}      102: {1,2,7}
   5: {3}        37: {12}       70: {1,3,4}    103: {27}
   6: {1,2}      38: {1,8}      71: {20}       105: {2,3,4}
   7: {4}        39: {2,6}      73: {21}       106: {1,16}
  10: {1,3}      41: {13}       74: {1,12}     107: {28}
  11: {5}        42: {1,2,4}    77: {4,5}      109: {29}
  13: {6}        43: {14}       78: {1,2,6}    110: {1,3,5}
  14: {1,4}      46: {1,9}      79: {22}       111: {2,12}
  15: {2,3}      47: {15}       82: {1,13}     113: {30}
  17: {7}        51: {2,7}      83: {23}       114: {1,2,8}
  19: {8}        53: {16}       85: {3,7}      115: {3,9}
  21: {2,4}      55: {3,5}      86: {1,14}     118: {1,17}
  22: {1,5}      57: {2,8}      87: {2,10}     119: {4,7}
  23: {9}        58: {1,10}     89: {24}       122: {1,18}
  26: {1,6}      59: {17}       91: {4,6}      123: {2,13}
  29: {10}       61: {18}       93: {2,11}     127: {31}
  30: {1,2,3}    62: {1,11}     94: {1,15}     129: {2,14}
  31: {11}       65: {3,6}      95: {3,8}      130: {1,3,6}

Row sums are A319246.
Row lengths are A072047.
Same as A319247 with rows reversed.
Composition of A000720 and A265668.
Looking at all numbers instead of just squarefree numbers gives A112798.
Cf. A000009, A005117, A048672, A056239, A299755, A302590.

Table[PrimePi/@First/@If[k==1,{},FactorInteger[k]],{k,Select[Range[30],SquareFreeQ]}]

A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

Original entry on oeis.org

1, 2, 6, 30, 180, 1260, 12600, 138600, 1801800, 25225200, 378378000, 6432426000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 33765373585944000, 979195833992376000, 29375875019771280000

Offset: 1

Leroy Quet, Oct 07 2005

nonn

Do all terms belong to A242031 (weakly decreasing prime signature)? - Gus Wiseman, May 14 2021

			Since the first 6 squarefree positive integers are 1, 2, 3, 5, 6, 7, the 6th term of the sequence is 1*2*3*5*6*7 = 1260.
From _Gus Wiseman_, May 14 2021: (Start)
The sequence of terms together with their prime signatures begins:
             1: ()
             2: (1)
             6: (1,1)
            30: (1,1,1)
           180: (2,2,1)
          1260: (2,2,1,1)
         12600: (3,2,2,1)
        138600: (3,2,2,1,1)
       1801800: (3,2,2,1,1,1)
      25225200: (4,2,2,2,1,1)
     378378000: (4,3,3,2,1,1)
    6432426000: (4,3,3,2,1,1,1)
  122216094000: (4,3,3,2,1,1,1,1)
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..417

A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A072047 applies Omega to each squarefree number.
A246867 groups squarefree numbers by Heinz weight (row sums: A147655).
A261144 groups squarefree numbers by smoothness (row sums: A054640).
A319246 gives the sum of prime indices of each squarefree number.
A329631 lists prime indices of squarefree numbers (reversed: A319247).
Cf. A001221, A002110, A014466, A070826, A338901, A339360.

Rest[FoldList[Times,1,Select[Range[40],SquareFreeQ]]] (* Harvey P. Dale, Jun 14 2011 *)

m=30;k=1;for(n=1,m,if(issquarefree(n),print1(k=k*n,",")))

More terms from Klaus Brockhaus, Oct 08 2005

A339360 Sum of all squarefree numbers with greatest prime factor prime(n).

Original entry on oeis.org

1, 2, 9, 60, 504, 6336, 89856, 1645056, 33094656, 801239040, 24246190080, 777550233600, 29697402470400, 1250501433753600, 55083063155097600, 2649111037319577600, 143390180403000115200, 8619643674791667302400, 534710099148093259776000, 36412881178052121329664000

Offset: 0

Gus Wiseman, Dec 04 2020

nonn

			The initial terms are:
   1 = 1,
   2 = 2,
   9 = 3 + 6,
  60 = 5 + 10 + 15 + 30.

Robert Israel, Table of n, a(n) for n = 0..349

A010036 takes prime indices here to binary indices, row sums of A209862.
A048672 takes prime indices to binary indices in squarefree numbers.
A054640 divides the n-th term by prime(n), row sums of A261144.
A072047 counts prime factors of squarefree numbers.
A339194 is the restriction to semiprimes, row sums of A339116.
A339195 has this as row sums.
A002110 lists primorials.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A056239 is the sum of prime indices of n (Heinz weight).
A246867 groups squarefree numbers by weight, with row sums A147655.
A319246 is the sum of prime indices of the n-th squarefree number.
A319247 lists reversed prime indices of squarefree numbers.
A329631 lists prime indices of squarefree numbers.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014466, A098350, A168472, A282935, A302590, A326878, A338901.

f:= proc(n) local i;
  `if`(n=0, 1, ithprime(n)) *mul(1+ithprime(i),i=1..n-1)
end proc:
map(f, [$0..20]); # Robert Israel, Dec 08 2020

Table[Sum[Times@@Prime/@stn,{stn,Select[Subsets[Range[n]],MemberQ[#,n]&]}],{n,10}]

For n >= 1, a(n) = A054640(n-1) * prime(n).

a(0)=1 prepended by Alois P. Heinz, Jan 08 2025

cp's OEIS Frontend

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A319247 Irregular triangle whose n-th row lists the strict integer partition whose Heinz number is the n-th squarefree number.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A329631 Irregular triangle read by rows where row n lists the prime indices of the n-th squarefree number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A339360 Sum of all squarefree numbers with greatest prime factor prime(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions