A246867 -id:A246867 - cp's OEIS frontend

A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n).

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128, 19, 34, 39, 49, 52, 55, 63, 66, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 256

Offset: 0

Alois P. Heinz, Aug 08 2012

The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1-6, 9, 10, 14, 15, 21, 22, 33, 49, 1095199, ... and inverse permutation A215501.
Number m is positioned in row n = A056239(m).  The number of different values m, such that both m and m+1 occur in row n is A088850(n).  A215369 lists all values m, such that both m and m+1 are in the same row.
The power prime(i)^j of the i-th prime is in row i*j for j in {0,1,2, ... }.
Column k=2 contains the even semiprimes A100484, where 10 and 22 are replaced by the odd semiprimes 9 and 21, respectively.
This triangle is related to the triangle A145518, see in both triangles the first column, the right border, the second right border and the row sums. - Omar E. Pol, May 18 2015

			The partitions of n=3 are {[3], [2,1], [1,1,1]}, encodings give {prime(3), prime(2)*prime(1), prime(1)^3} = {5, 3*2, 2^3} => row 3 = [5, 6, 8].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3,  4;
   5,  6,  8;
   7,  9, 10, 12, 16;
  11, 14, 15, 18, 20, 24, 32;
  13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64;
  17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128;
  ...
Corresponding triangle of integer partitions begins:
  ();
  1;
  2, 11;
  3, 21, 111;
  4, 22, 31, 211, 1111;
  5, 41, 32, 221, 311, 2111, 11111;
  6, 42, 51, 33, 222, 411, 321, 2211, 3111, 21111, 111111;
  7, 61, 52, 43, 421, 511, 322, 331, 2221, 4111, 3211, 22111, 31111, 211111, 1111111;  - _Gus Wiseman_, Dec 12 2016

Column k=1 gives: A008578(n+1).
Last elements of rows give: A000079.
Second to last elements of rows give: A007283(n-2) for n>1.
Row sums give: A145519.
Row lengths are: A000041.
Cf. A129129 (with row elements using order of A080577).
LCM of terms in row n gives A138534(n).
Cf. A000027, A000040, A056239, A063008, A088850, A100484, A215501.
Cf. A112798, A246867 (the same for partitions into distinct parts).
Cf. A324939, A377852, A378175.

b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
       [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> sort(b(n, n))[]:
seq(T(n), n=0..10);
# (2nd Maple program)
with(combinat): A := proc (n) local P, A, i: P := partition(n): A := {}; for i to nops(P) do A := `union`(A, {mul(ithprime(P[i][j]), j = 1 .. nops(P[i]))}) end do: A end proc; # the command A(m) yields row m. # Emeric Deutsch, Jan 23 2016
# (3rd Maple program)
q:= 7: S[0] := {1}: for m to q do S[m] := `union`(seq(map(proc (f) options operator, arrow: ithprime(j)*f end proc, S[m-j]), j = 1 .. m)) end do; # for a given positive integer q, the program yields rows 0, 1, 2,...,q. # Emeric Deutsch, Jan 23 2016

b[n_, i_] := b[n, i] = If[n == 0 || i<2, {2^n}, Table[Function[#*Prime[i]^j] /@ b[n - i*j, i-1], {j, 0, n/i}] // Flatten]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)
nn=7;HeinzPartition[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]//Reverse];
Take[GatherBy[Range[2^nn],Composition[Total,HeinzPartition]],nn+1] (* Gus Wiseman, Dec 12 2016 *)
Table[Map[Times @@ Prime@ # &, IntegerPartitions[n]], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jul 12 2017 *)

\\ From M. F. Hasler, Dec 06 2016 (Start)
A215366_row(n)=vecsort([vecprod([prime(p)|p<-P])|P<-partitions(n)]) \\ bug fix & syntax update by M. F. Hasler, Oct 20 2023
A215366_vec(N)=concat(apply(A215366_row,[0..N])) \\ "flattened" rows 0..N (End)

Recurrence relation, explained for the set S(4) of entries in row 4: multiply the entries of S(3) by 2 (= 1st prime), multiply the entries of S(2) by 3 (= 2nd prime), multiply the entries of S(1) by 5 (= 3rd prime), multiply the entries of S(0) by 7 (= 4th prime); take the union of all the obtained products. The 3rd Maple program is based on this recurrence relation. - Emeric Deutsch, Jan 23 2016

A299755 Triangle read by rows in which row n is the strict integer partition with FDH number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 18 2018

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

			Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9) (10) (6,1) (11) (4,3) (5,2) (7,1) (12) (3,2,1) (13) (8,1) (6,2) (5,3) (14) (4,2,1) (15).

Row lengths are A064547.

Cf. A005117, A050376, A112798, A213925, A246867, A296150, A299756, A299757, A299759.

FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Join@@Table[Reverse[FDfactor[n]/.FDrules],{n,nn}]

A299757 Weight of the strict integer partition with FDH number n.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 5, 8, 6, 6, 9, 10, 7, 11, 7, 7, 8, 12, 6, 13, 9, 8, 8, 14, 7, 15, 10, 9, 11, 9, 9, 16, 12, 10, 8, 17, 8, 18, 10, 10, 13, 19, 11, 20, 14, 12, 11, 21, 9, 11, 9, 13, 15, 22, 9, 23, 16, 11, 12, 12, 10, 24, 13, 14, 10, 25, 10, 26, 17

Offset: 1

Gus Wiseman, Feb 18 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

In analogy with the Heinz number correspondence between integer partitions and positive integers (see A056239), FDH numbers give a correspondence between strict integer partitions and positive integers.

			Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9).

Cf. A004111, A050376, A056239, A061775, A064547, A106400, A213925, A215366, A246867, A279065, A279614, A299090, A299755, A299756, A299758, A299759.

FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[FDfactor[n]/.FDrules],{n,nn}]

A147655 a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).

Original entry on oeis.org

1, 2, 3, 11, 17, 40, 86, 153, 283, 547, 1069, 1737, 3238, 5340, 9574, 17251, 27897, 45845, 78601, 126725, 207153, 353435, 550422, 881454, 1393870, 2239938, 3473133, 5546789, 8762663, 13341967, 20676253, 31774563, 48248485, 74174759, 111904363, 170184798

Offset: 0

Neil Fernandez, Nov 09 2008

nonn

Sum of all squarefree numbers whose prime indices sum to n. 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. - Gus Wiseman, May 09 2019

			Form a product from the primes: (1 + 2*x) * (1 + 3*x^2) * (1 + 5*x^3) * ...* (1 + prime(n)*x^n) * ... Multiplying out gives 1 + 2*x + 3*x^2 + 11*x^3 + ..., so the sequence begins 1, 2, 3, 11, ....
From _Petros Hadjicostas_, Apr 10 2020: (Start)
Let f(m) = prime(m). Using the strict partitions of n (see A000009), we get:
a(1) = f(1) = 2,
a(2) = f(2) = 3,
a(3) = f(3) + f(1)*f(2) = 5 + 2*3 = 11,
a(4) = f(4) + f(1)*f(3) = 7 + 2*5 = 17,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 11 + 2*7 + 3*5 = 40,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 13 + 2*11 + 3*7 + 2*3*5 = 86,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 17 + 2*13 + 3*11 + 5*7 + 2*3*7 = 153. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..9000 (first 1001 terms from Harvey P. Dale)

Row sums of A246867 and A258323.

Cf. A000009,A005117, A015723, A022629, A056239, A066189, A112798, A145519, A147541, A325504, A325506, A325537.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 05 2014

nn=40;Take[Rest[CoefficientList[Expand[Times@@Table[1+Prime[n]x^n,{n,nn}]],x]],nn] (* Harvey P. Dale, Jul 01 2012 *)

a(n) = [x^n] Product_{k>=1} 1+prime(k)*x^k. - Alois P. Heinz, Sep 05 2014

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = prime(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 10 2020

More terms from Harvey P. Dale, Jul 01 2012
a(0)=1 inserted by Alois P. Heinz, Sep 05 2014
Name edited by Petros Hadjicostas, Apr 10 2020

A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 24, 19, 21, 25, 22, 26, 28, 23, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 41, 49, 38, 42, 50, 44, 52, 56, 39, 43, 51, 45, 53, 57, 46, 54, 58, 60, 47, 55, 59, 61, 62, 63, 64, 65, 66, 68, 72, 80, 96, 67, 69, 73, 81, 97, 70, 74, 82, 98, 76, 84, 100, 88, 104, 112, 71, 75, 83

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

Conjecture: For all n, a(A054429(n)) = A054429(a(n)), i.e. A054429 acts as a homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860.
From Gus Wiseman, Aug 24 2021: (Start)
As a triangle with row lengths 2^n, T(n,k) for n > 0 appears (verified up to n = 2^15) to be the unique nonnegative integer whose binary indices are the k-th subset of {1..n} containing n. Here, a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion, and sets are sorted first by length, then lexicographically. For example, the triangle begins:
   1
   2  3
   4  5  6  7
   8  9 10 12 11 13 14 15
  16 17 18 20 24 19 21 25 22 26 28 23 27 29 30 31
Mathematica: Table[Total[2^(Append[#,n]-1)]&/@Subsets[Range[n-1]],{n,5}]
Row lengths are A000079 (shifted right). Also Column k = 1.
Row sums are A010036.
Using reverse-lexicographic order gives A059893.
Using lexicographic order gives A059894.
Taking binary indices to prime indices gives A339195 (or A019565).
The ordering of sets is A344084.
A version using Heinz numbers is A344085.
(End)

			From _Gus Wiseman_, Aug 24 2021: (Start)
The terms, their binary expansions, and their binary indices begin:
   0:      ~ {}
   1:    1 ~ {1}
   2:   10 ~ {2}
   3:   11 ~ {1,2}
   4:  100 ~ {3}
   5:  101 ~ {1,3}
   6:  110 ~ {2,3}
   7:  111 ~ {1,2,3}
   8: 1000 ~ {4}
   9: 1001 ~ {1,4}
  10: 1010 ~ {2,4}
  12: 1100 ~ {3,4}
  11: 1011 ~ {1,2,4}
  13: 1101 ~ {1,3,4}
  14: 1110 ~ {2,3,4}
  15: 1111 ~ {1,2,3,4}
(End)

Antti Karttunen, Table of n, a(n) for n = 0..32767
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A209861. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

Cf. A010036, A026793, A048793, A111059, A147655, A246688, A246867, A261144, A272020, A294648, A339360.

a(n) = A209859(A036044(A209641(n))) = A209859(A056539(A209641(n))).

A258323 Sum T(n,k) over all partitions lambda of n into k distinct parts of Product_{i:lambda} prime(i); triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

Original entry on oeis.org

1, 0, 2, 0, 3, 0, 5, 6, 0, 7, 10, 0, 11, 29, 0, 13, 43, 30, 0, 17, 94, 42, 0, 19, 128, 136, 0, 23, 231, 293, 0, 29, 279, 551, 210, 0, 31, 484, 892, 330, 0, 37, 584, 1765, 852, 0, 41, 903, 2570, 1826, 0, 43, 1051, 4273, 4207, 0, 47, 1552, 6747, 6595, 2310

Offset: 0

Alois P. Heinz, May 26 2015

			T(6,2) = 43 because the partitions of 6 into 2 distinct parts are {[5,1], [4,2]} and prime(5)*prime(1) + prime(4)*prime(2) = 11*2 + 7*3 = 22 + 21 = 43.
Triangle T(n,k) begins:
  1
  0,  2;
  0,  3;
  0,  5,   6;
  0,  7,  10;
  0, 11,  29;
  0, 13,  43,  30;
  0, 17,  94,  42;
  0, 19, 128, 136;
  0, 23, 231, 293;
  0, 29, 279, 551, 210;

Alois P. Heinz, Rows n = 0..500, flattened

Columns k=0-10 give: A000007, A000040, A025129(n+1), A258358, A258359, A258360, A258361, A258362, A258363, A258364, A258365.
Row sums give A147655.
T(n*(n+1)/2,n) = A002110(n).
T(n^2,n) = A321267(n).
Cf. A000217, A003056, A145518, A246867.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, expand(
      add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..min(1, n/i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2)):
seq(T(n), n=0..20);

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Expand[Sum[g[n-i*j, i-1] * (Prime[i]*x)^j, {j, 0, Min[1, n/i]}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)

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

Original entry on oeis.org

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

A325504 Product of products of parts over all strict integer partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 120, 1440, 40320, 1209600, 1567641600, 2633637888000, 13905608048640000, 5046067048690483200000, 5289893008483207348224000000, 1266933607446134946465526579200000000, 99304891373531545064656621572980736000000000000

Offset: 0

Gus Wiseman, May 07 2019

nonn

			The strict partitions of 5 are {(5), (4,1), (3,2)} with product a(5) = 5*4*1*3*2 = 120.
The sequence of terms together with their prime indices begins:
              1: {}
              1: {}
              2: {1}
              6: {1,2}
             12: {1,1,2}
            120: {1,1,1,2,3}
           1440: {1,1,1,1,1,2,2,3}
          40320: {1,1,1,1,1,1,1,2,2,3,4}
        1209600: {1,1,1,1,1,1,1,1,2,2,2,3,3,4}
     1567641600: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4}
  2633637888000: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4}

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A000009, A006128, A007870 (non-strict version), A015723, A022629 (sum of products of parts), A066186, A066189, A066633, A246867, A325505, A325506, A325512, A325513, A325515.

a:= n-> mul(i, i=map(x-> x[], select(x->
        nops(x)=nops({x[]}), combinat[partition](n)))):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0, 1], ((f, g)->
     [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 03 2021

Table[Times@@Join@@Select[IntegerPartitions[n],UnsameQ@@#&],{n,0,10}]

A001222(a(n)) = A325515(n).

a(n) = A003963(A325506(n)).

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

cp's OEIS Frontend

A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n).

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A299755 Triangle read by rows in which row n is the strict integer partition with FDH number n.

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A299757 Weight of the strict integer partition with FDH number n.

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A147655 a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).

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A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).

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A258323 Sum T(n,k) over all partitions lambda of n into k distinct parts of Product_{i:lambda} prime(i); triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

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

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

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