A146292 -id:A146292 - cp's OEIS frontend

A146291 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001222(n)), giving the number of divisors of n with m prime factors (counted with multiplicity).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2

Offset: 1

Matthew Vandermast, Nov 11 2008

All rows are palindromic. T(n,0) = T(n,A001222(n)) = 1.
Two numbers have identical rows in the table if and only if they have the same prime signature.
If n is a perfect square then Sum_{even m} T(n,m) = 1 + Sum_{odd m} T(n,m), otherwise Sum_{even m} T(n,m) = Sum_{odd m} T(n,m). - Geoffrey Critzer, Feb 08 2015

			Rows begin:
1;
1, 1;
1, 1;
1, 1, 1;
1, 1;
1, 2, 1;
1, 1;
1, 1, 1, 1;
1, 1, 1;
1, 2, 1;
...
12 has 1 divisor with 0 total prime factors (1), 2 with 1 (2 and 3), 2 with 2 (4 and 6) and 1 with 3 (12), for a total of 6. The 12th row of the table therefore reads (1, 2, 2, 1). These are the positive coefficients of the polynomial 1 + 2k + 2k^2 + (1)k^3 = (1 + k + k^2)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1).

Alois P. Heinz, Rows n = 1..2500, flattened
Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Roundness
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Row sums equal A000005(n). T(n,1) = A001221(n) for n>1.
Row n of A007318 is identical to row A002110(n) of this table and also identical to the row for any squarefree number with n prime factors.
Cf. A146292, A146289, A146290.

with(numtheory):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
         add(x^bigomega(d), d=divisors(n))):
seq(T(n), n=1..100);  # Alois P. Heinz, Feb 25 2015

Join[{{1}},
Table[nn = DivisorSigma[0, n];
  CoefficientList[
   Series[Product[(1 - x^i)/(1 - x), {i,
FactorInteger[n][[All, 2]] + 1}], {x, 0, nn}], x], {n, 2, 100}]] (* Geoffrey Critzer, Jan 01 2015 *)

If the canonical factorization of n into prime powers is the product of p^e(p), then T(n,m) is the coefficient of k^m in the polynomial expansion of Product_p (sum_{i=0..e} k^i).

A146289 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001221(n)), giving the number of divisors of n with m distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 3, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 4, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 3, 1, 1, 1, 3, 3

Offset: 1

Matthew Vandermast, Nov 11 2008

The formula used in obtaining the n-th row (see below) also gives the number of divisors of the k-th power of n.
Two numbers have identical rows in the table if and only if they have the same prime signature.
T(n,0)=1.

			Rows begin:
1;
1,1;
1,1;
1,2;
1,1;
1,2,1;
1,1;
1,3;
1,2;
1,2,1;
...
12 has 1 divisor with 0 distinct prime factors (1); 3 with 1 (2, 3 and 4); and 2 with 2 (6 and 12), for a total of 6. The 12th row of the table therefore reads (1, 3, 2). These are the positive coefficients of the polynomial equation 1 + 3k + 2k^2 = (1 + 2k)(1 + k), derived from the prime factorization of 12 (namely, 2^2*3^1).

G. C. Greubel, Table of n, a(n) for the first 500 rows, flattened
Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

Row sums equal A000005(n).
T(n, 1) = A001222(n) for n>1. T(n, A001221(n)) = A005361(n).
Row n of A007318 is identical to row A002110(n) of this table and also identical to the row for any squarefree number with n prime factors.
Cf. A146290. Also cf. A146291, A146292.

f:= proc(n)
   local F,G,f,t,k;
   F:= ifactors(n)[2];
   G:= mul(1+f[2]*t, f= F);
   seq(coeff(G,t,k),k=0..nops(F));
end proc:
seq(f(n),n=1..100); # Robert Israel, Feb 10 2015

Join[{{1}}, Table[nn = DivisorSigma[0, n];CoefficientList[Series[Product[1 + i x, {i, FactorInteger[n][[All, 2]]}], {x, 0,nn}], x], {n, 2, 100}]] // Grid (* Geoffrey Critzer, Feb 09 2015 *)

tabf(nn) = {for (n=1, nn, vd = divisors(n); vo = vector(#vd, k, omega(vd[k])); for (k=0, vecmax(vo), print1(#select(x->x==k, vo), ", ");); print(););} \\ Michel Marcus, Apr 22 2017

If the canonical factorization of n into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + e(p) k).

A181555 a(n) = A002110(n)^n.

Original entry on oeis.org

1, 2, 36, 27000, 1944810000, 65774855015100000, 733384949590939374729000000, 9037114296609938214167920266348510000000, 78354300210436852307898467208663359164858967744100000000

Offset: 0

Matthew Vandermast, Oct 31 2010

For n>0, a(n)= first counting number whose prime signature consists of n repeated n times (cf. A002024). Subsequence of A025487.

			a(4) = 1944810000 = 210^4 = 2^4 * 3^4 * 5^4 * 7^4.

Amiram Eldar, Table of n, a(n) for n = 0..26

A061742(n) = A002110(n)^2. See also A006939, A066120, A166475, A167448.
A000005(a(n)) = A000169(n). The divisors of a(n) appear as the first A000169(n) terms of A178479, with A178479(A000169(n)) = a(n).
A071207(n, k) gives the number of divisors of n with (n-k) distinct prime factors, A181567(n, k) gives the number of divisors of n with k prime factors counted with multiplicity.
Cf. A001221, A001222, A079474, A146290, A146292.

a[0] = 1; a[n_] := Product[Prime[i], {i, 1, n}]^n; Array[a, 9, 0] (* Amiram Eldar, Aug 08 2019 *)

a(n) = A079474(2n,n). - Alois P. Heinz, Aug 22 2019

A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 4, 1, 4, 3, 1, 3, 3, 1, 1, 5, 1, 4, 4, 1, 5, 4, 1, 4, 5, 2, 1, 6, 1, 5, 6, 1, 6, 5, 1, 5, 7, 3, 1, 7, 1, 6, 8, 1, 5, 8, 4, 1, 7, 6, 1, 4, 6, 4, 1, 1, 6, 9, 1, 6, 9, 4, 1, 8, 1, 7, 10, 1, 6, 11, 6, 1, 8, 7, 1, 5, 9, 7, 2, 1, 7, 12, 1, 7, 11, 5, 1, 9, 1, 8, 12, 1, 7, 14

Offset: 1

Matthew Vandermast, Nov 11 2008

The formula used in obtaining the A025487(n)th row (see below) also gives the number of divisors of the k-th power of A025487(n).
Every row that appears in A146289 appears exactly once in the table. Rows appear in order of first appearance in A146289.
T(n,0)=1.

			Rows begin:
  1;
  1,1;
  1,2;
  1,2,1;
  1,3;
  1,3,2;
  1,4;
  1,4,3;...
36's 9 divisors include 1 divisor with 0 distinct prime factors (1); 4 with 1 (2, 3, 4 and 9); and 4 with 2 (6, 12, 18 and 36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 4, 4). These are the positive coefficients of the polynomial equation 1 + 4k + 4k^2 = (1 + 2k)(1 + 2k), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of distinct prime factors of n, see A001221.
Row sums equal A146288(n). T(n, 1)=A036041(n) for n>1. T(n, A061394(n))=A052306(n).
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146289. Also cf. A146291, A146292.

If A025487(n)'s canonical factorization into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + ek).

A146288 Number of divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 8, 6, 9, 10, 12, 7, 12, 12, 16, 8, 15, 18, 14, 16, 16, 20, 9, 18, 24, 16, 24, 20, 24, 10, 21, 30, 18, 32, 24, 27, 28, 11, 32, 24, 36, 25, 36, 20, 40, 28, 36, 32, 12, 40, 27, 32, 48, 30, 42, 22, 48, 32, 45, 36, 13, 48, 30, 48, 60, 35, 48, 48, 24, 54, 50, 56

Offset: 1

Matthew Vandermast, Nov 11 2008

nonn

			a(4) = 4 because 4 positive integers divide evenly into A025487(4) = 6: 1, 2, 3 and 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

a(n) = sum of the n-th row of A146290, A146292.
A rearrangement of A080444.
Cf. A000005 (number of divisors), A025487.

a146288 = a000005 . a025487  -- Reinhard Zumkeller, Sep 17 2014

s = {1}; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], AppendTo[s, DivisorSigma[0, n]]], {n, 2, 10000}]; s (* Amiram Eldar, Aug 05 2024 *)

a(n) = A000005(A025487(n)).

A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780, 780, 735, 651, 540, 420, 305, 205, 126, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 462, 786, 1251

Offset: 0

Matthew Vandermast, Oct 31 2010

In each row n>=0, k takes values from 0 to n^2 inclusive. Row sums equal A000169(n+1). All rows are palindromic. Row n is also row n of the (n+1)-nomial array (e.g., row 1 is also row 1 of A007318).
T(n,k) gives the number of divisors of A181555(n) with k prime factors counted with multiplicity. See also A001222, A071207, A146291, A146292.
T(n,k) is the number of size k submultisets of the so-called regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 0 < i < n). - Thomas Wieder, Dec 28 2013

			Rows begin:
1;
1,1;
1,2,3,2,1;
1,3,6,10,12,12,10,6,3,1;...
T(n=3,k=4) = 12 because we have 12 submultisets (without regard of the order of elements) of size k=4 for the regular multiset (n=3) {1, 1, 1, 2, 2, 2, 3, 3, 3}: {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 3, 3}, {1, 2, 2, 2}, {1, 2, 2, 3}, {1, 2, 3, 3}, {1, 3, 3, 3}, {2, 2, 2, 3}, {2, 2, 3, 3}, {2, 3, 3, 3}.

Alois P. Heinz, Rows n = 0..32, flattened
Anonymous, Polynomial calculator
Thomas Wieder, A181567 as Excel table
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

A163181 gives row n of n-nomial array. See also A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890.

b:= proc(n, k, i) option remember; `if`(k=0, 1,
     `if`(i<1, 0, add(b(n, k-j, i-1), j=0..n)))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n^2), n=0..8); # Alois P. Heinz, Jul 04 2016

row[n_] := CoefficientList[((x^(n+1) - 1)/(x-1))^n + O[x]^(n^2+1), x]; Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Apr 06 2017 *)

cp's OEIS Frontend

A146291 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001222(n)), giving the number of divisors of n with m prime factors (counted with multiplicity).

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A146289 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A001221(n)), giving the number of divisors of n with m distinct prime factors.

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A181555 a(n) = A002110(n)^n.

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A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

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A146288 Number of divisors of the n-th prime signature number (A025487(n)).

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A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.

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