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.

Showing 1-3 of 3 results.

A107329 Triangle read by rows: T(n,k) gives number of partitions of k, (k=1..n) into the prime factors of n, for n>=1.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Wouter Meeussen, May 22 2005

Keywords

Comments

T(n,n) equals A066882(n).

Examples

			T(30,12)=5 counting [2,2,2,2,2,2], [2,2,2,3,3], [3,3,3,3], [2,2,3,5] and [2,5,5].
Triangle begins:
  {0},
  {0, 1},
  {0, 0, 1},
  {0, 1, 0, 1},
  {0, 0, 0, 0, 1},
  {0, 1, 1, 1, 1, 2},
  {0, 0, 0, 0, 0, 0, 1},
  {0, 1, 0, 1, 0, 1, 0, 1},
  {0, 0, 1, 0, 0, 1, 0, 0, 1},
  ...

Links

Crossrefs

Cf. A066882.
Row sums +1 give A092976.

Programs

  • Maple
    with(numtheory):
T:= proc(n) local b, l; l:= sort([factorset(n)[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
            b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      seq(b(k, nops(l)), k=1..n)
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Oct 28 2021
  • Mathematica
    Table[Rest@CoefficientList[Series[1/Times @@ ((1-x^#)& /@ (First /@ FactorInteger[n])), {x, 0, n}], x], {n, 2, 24}]

Formula

T(n,k) is coefficient of x^k in 1/Product(1-x^p_i) with p_i the prime factors of n.

Extensions

T(1,1) = 0 prepended by Michel Marcus, Oct 28 2021

A219208 Number of distinct products of all parts of all partitions of n into distinct divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 7, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 26, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 23, 1
Offset: 0

Views

Author

Alois P. Heinz, Nov 14 2012

Keywords

Comments

a(p) = 1 for p in A000040 (prime numbers).
a(n) = 1 for n in A006037 (weird numbers).
a(n) = 1 for n in A005100 (deficient numbers).
a(n) = 1 for n in A125493 (composite deficient numbers).
a(n) <= 2 for n in A000396 (perfect numbers).
a(n) >= 2 for n > 6 and n in A005835 (semiperfect numbers).

Examples

			a(0) = 1: the empty product.
a(p) = 1 for any prime p: [p]-> p.
a(6) = 1: {[1,2,3], [6]}-> 6.
a(12) = 3, because all 3 partitions of 12 into distinct divisors of 12 have different products: [1,2,3,6]-> 36, [2,4,6]-> 48, [12]-> 12. a(18) = 3: [1,2,6,9]-> 108, [3,6,9]-> 162, [18]-> 18.
a(20) = 2: [1,4,5,10]-> 200, [20]-> 20.
a(28) = 2: [1,2,4,7,14]-> 784, [28]-> 28.
a(36) = 7: [2,3,4,6,9,12]-> 15552, [2,3,4,9,18]-> 3888, [1,2,6,9,18]-> 1944, [3,6,9,18]-> 2916, {[1,2,3,12,18], [6,12,18]}-> 1296, [2,4,12,18]-> 1728, [36]-> 36.
a(84) = 23: 84, 16464, 28224, 49392, 65856, 74088, 84672, 86436, 98784, 127008, 148176, 190512, 254016, 444528, 592704, 889056, 1016064, 1185408, 1382976, 1778112, 2370816, 4148928, 7112448.

Links

Crossrefs

Maximal products are in A219209.
Cf. A033630, A092976.

Programs

  • Maple
    a:= proc(n) local b, l;
      l:= sort([numtheory[divisors](n)[]]);
      b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
            {b(n, i-1)[], `if`(l[i]>n, {}, map(x-> x*l[i],
            b(n-l[i], i-1)))[]}))
          end; forget(b);
      nops(b(n, nops(l)))
    end:
seq(a(n), n=0..120);
  • Mathematica
    a[n_] := a[n] = Module[{b, l}, l = Divisors[n]; b[m_, i_] := b[m, i] = If[m == 0, {1}, If[i<1, {}, Union[b[m, i-1], If[l[[i]]>m, {}, (#*l[[i]]&) /@ b[m-l[[i]], i-1]]]]]; Length[b[n, Length[l]]]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)

A219209 Maximal product of all parts of a partition of n into distinct divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 48, 13, 14, 15, 16, 17, 162, 19, 200, 21, 22, 23, 1152, 25, 26, 27, 784, 29, 1350, 31, 32, 33, 34, 35, 15552, 37, 38, 39, 6400, 41, 2058, 43, 44, 45, 46, 47, 73728, 49, 50, 51, 52, 53, 8748, 55, 25088, 57, 58, 59, 864000
Offset: 0

Views

Author

Alois P. Heinz, Nov 14 2012

Keywords

Examples

			a(0) = 1: the empty product.
a(p) = p for any prime p: [p]-> p.
a(12) = 48: [2,4,6]-> 48.
a(20) = 200: [1,4,5,10]-> 200.
a(24) = 1152: [1,2,3,4,6,8]-> 1152.

Links

Crossrefs

The number of distinct products are in A219208.
Cf. A033630, A092976.

Programs

  • Maple
    a:= proc(n) local b, l;
      l:= sort([numtheory[divisors](n)[]]);
      b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
            max(b(n, i-1), `if`(l[i]>n, 0, l[i] *b(n-l[i], i-1)))))
          end; forget(b);
      b(n, nops(l))
    end:
seq(a(n), n=0..80);
  • Mathematica
    a[n_] := a[n] = Module[{b, l}, l = Divisors[n]; b[m_, i_] := b[m, i] = If[m == 0, 1, If[i<1, 0, Max[b[m, i-1], If[l[[i]]>m, 0, l[[i]]*b[m-l[[i]], i-1] ]]]]; b[n, Length[l]]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)
Showing 1-3 of 3 results.

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