A052146 -id:A052146 - cp's OEIS frontend

A121431 Number of subpartitions of partition P=[0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,...] (A052146).

1, 1, 1, 2, 3, 4, 9, 15, 22, 30, 69, 118, 178, 250, 335, 769, 1317, 1995, 2820, 3810, 4984, 11346, 19311, 29126, 41061, 55410, 72492, 92652, 208914, 352636, 528097, 740035, 993678, 1294776, 1649634, 2065146, 4613976, 7722840, 11476963, 15971180

Offset: 0

Paul D. Hanna, Jul 30 2006

nonn

See A115728 for the definition of subpartitions of a partition.

			The g.f. may be illustrated by:
1/(1-x) = (1 + 1*x)*(1-x)^0 + (x^2 + 2*x^3 + 3*x^4)*(1-x)^1 +
(4*x^5 + 9*x^6 + 15*x^7 + 22*x^8)*(1-x)^2 +
(30*x^9 + 69*x^10 + 118*x^11 + 178*x^12 + 250*x^13)*(1-x)^3 +
(335*x^14 + 769*x^15 + 1317*x^16 + 1995*x^17 + 2820*x^18 + 3810*x^19)*(1-x)^4 +...
When the sequence is put in the form of a triangle:
1, 1,
1, 2, 3,
4, 9, 15, 22,
30, 69, 118, 178, 250,
335, 769, 1317, 1995, 2820, 3810,
4984, 11346, 19311, 29126, 41061, 55410, 72492,
92652, 208914, 352636, 528097, 740035, 993678, 1294776, ...
then the columns of this triangle form column 1 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121426 as follows.
Column 1 of successive powers of matrix H begin:
H^1: [1,1,4,30,335,4984,92652,2065146,53636520,...];
H^2: [1,2,9,69,769,11346,208914,4613976,118840164,...];
H^3: 1, [3,15,118,1317,19311,352636,7722840,197354133,...];
H^4: 1,4, [22,178,1995,29126,528097,11476963,291124693,...];
H^5: 1,5,30, [250,2820,41061,740035,15971180,402319275,...];
H^6: 1,6,39,335, [3810,55410,993678,21310710,533345745,...];
H^7: 1,7,49,434,4984, [72492,1294776,27611970,686872893,...];
H^8: 1,8,60,548,6362,92652, [1649634,35003430,865852191,...];
H^9: 1,9,72,678,7965,116262,2065146, [43626510,1073540871,...];
the terms enclosed in brackets form this sequence.

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121426, A121427; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121432, A121433.

{a(n)=local(A); if(n==0,1,A=x+x*O(x^n); for(k=0, n, A+=polcoeff(A, k)*x^k*(1-(1-x)^( (sqrtint(8*k+9)+1)\2 - 1 ) )); polcoeff(A, n))}

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^A052146(n).

A241719 Number T(n,k) of compositions of n into distinct parts with exactly k descents; triangle T(n,k), n>=0, 0<=k<=max(floor((sqrt(1+8*n)-3)/2),0), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 4, 6, 1, 5, 7, 1, 6, 11, 2, 8, 16, 3, 10, 31, 15, 1, 12, 36, 16, 1, 15, 55, 29, 2, 18, 71, 41, 3, 22, 101, 65, 5, 27, 147, 144, 32, 1, 32, 188, 179, 35, 1, 38, 245, 269, 63, 2, 46, 327, 382, 93, 3, 54, 421, 549, 148, 5, 64, 540, 739, 205, 7

Offset: 0

Alois P. Heinz, Apr 27 2014

			T(6,0) = 4: [6], [2,4], [1,5], [1,2,3].
T(6,1) = 6: [5,1], [4,2], [3,1,2], [1,3,2], [2,1,3], [2,3,1].
T(6,2) = 1: [3,2,1].
T(7,0) = 5: [7], [3,4], [2,5], [1,6], [1,2,4].
T(7,1) = 7: [6,1], [4,3], [5,2], [2,1,4], [1,4,2], [2,4,1], [4,1,2].
T(7,2) = 1: [4,2,1].
Triangle T(n,k) begins:
00:   1;
01:   1;
02:   1;
03:   2,   1;
04:   2,   1;
05:   3,   2;
06:   4,   6,   1;
07:   5,   7,   1;
08:   6,  11,   2;
09:   8,  16,   3;
10:  10,  31,  15,  1;
11:  12,  36,  16,  1;
12:  15,  55,  29,  2;
13:  18,  71,  41,  3;
14:  22, 101,  65,  5;
15:  27, 147, 144, 32, 1;

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

Columns k=0-10 give: A000009, A241720, A241721, A241722, A241723, A241724, A241725, A241726, A241727, A241728, A241729.
Row sums give A032020.
T(A000217(k+1)-1,k-1) = A000041(k) for k>0.
Cf. A052146.

g:= proc(u, o) option remember; `if`(u+o=0, 1, expand(
      add(g(u+j-1, o-j)  , j=1..o)+
      add(g(u-j, o+j-1)*x, j=1..u)))
    end:
b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      `if`(n>m, 0, `if`(n=m, x^i,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
T:= n-> (p-> (q-> seq(coeff(q, x, i), i=0..degree(q)))(add(
         coeff(p, x, k)*g(0, k), k=0..degree(p))))(b(n$2)):
seq(T(n), n=0..20);

g[u_, o_] := g[u, o] = If[u+o == 0, 1, Expand[Sum[g[u+j-1, o-j], {j, 1, o}] + Sum[g[u-j, o+j-1]*x, {j, 1, u}]]]; b[n_, i_] := b[n, i] = Module[{m}, m = i*(i+1)/2; If[n>m, 0, If[n == m, x^i, Expand[b[n, i-1] + If[i>n, 0, x*b[n-i, i-1]]]]]]; T[n_] := Function [p, Function[q, Table[Coefficient[q, x, i], {i, 0, Exponent[q, x]}]][Sum[Coefficient[p, x, k]*g[0, k], {k, 0, Exponent[p, x]}]]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Apr 28 2014, after Alois P. Heinz *)

A238406 Number T(n,k) of partitions of n into k parts such that every i-th smallest part (counted with multiplicity) is different from i; triangle T(n,k), n>=0, 0<=k<=floor((sqrt(9+8*n)-3)/2) read by rows.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 4, 3, 0, 1, 4, 4, 0, 1, 5, 6, 0, 1, 5, 7, 0, 1, 6, 9, 1, 0, 1, 6, 11, 4, 0, 1, 7, 13, 7, 0, 1, 7, 15, 11, 0, 1, 8, 18, 15, 0, 1, 8, 20, 19, 0, 1, 9, 23, 25, 1, 0, 1, 9, 26, 30, 5

Offset: 0

Alois P. Heinz, Feb 26 2014

			T(10,1) = 1: [10].
T(10,2) = 4: [5,5], [4,6], [3,7], [2,8].
T(10,3) = 3: [3,3,4], [2,4,4], [2,3,5].
Triangle T(n,k) begins:
  1;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2;
  0, 1, 3;
  0, 1, 3, 1;
  0, 1, 4, 3;
  0, 1, 4, 4;
  0, 1, 5, 6;
  0, 1, 5, 7;
  0, 1, 6, 9, 1;
  ...

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

Columns k=0-10 give: A000007, A000012 (for n>1), A004526(n-2) (for n>4), A244239, A244240, A244241, A244242, A244243, A244244, A244245, A244246.
Row sums give A238394.
Cf. A052146.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand(
       x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n$2)):
seq(T(n), n=0..30);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p - Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]] ]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Max[0, Exponent[p, x]]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

A113402 Algebraic degree of cos(Pi/n) for constructible n-gons (A003401).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 512, 512, 512

Offset: 1

Eric W. Weisstein, Oct 28 2005

a(n) is always a power of 2.
It would appear that a(n) <= a(n+1) and that for a(n)=2^k, the count for k beginning with 0 is 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, ...; or that the count for k is k+2 for k > 0. - Robert G. Wilson v, Jul 31 2014
Apparently v_2(a(n)) = A052146(n-1) for n >= 2 where v_2 is the 2-adic valuation. - Joerg Arndt, Jul 29 2014 [incorrect for n >= 561, Joerg Arndt, Mar 03 2019]

Robert G. Wilson v, Table of n, a(n) for n = 1..1632
Eric Weisstein's World of Mathematics, Trigonometry Angles

Cf. A000217, A002024, A003401, A055034, A113401.

f[n_] := Exponent[MinimalPolynomial[Cos[Pi/n]][x], x]; Table[ f@ n, {n, Select[ Range@ 1300, IntegerQ[ Log[2, EulerPhi[#]]] &]}] (* Robert G. Wilson v, Jul 28 2014 *)
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 2500; t = Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}]; f[n_] := EulerPhi[ 2n]/2; f[1] = 1; f@# & /@ t (* Robert G. Wilson v, Jul 28 2014 *)

A385379 The maximum possible number of distinct composite prime powers (A246547) in the factorization of n into prime powers.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Jun 27 2025

Differs from A376679 at n = 1, 48, 72, 80, ... .
The factorization includes primes if n is not a powerful number (A001694) that is larger than 1.
a(n) depends only on the prime signature of n (A118914).

			         n | a(n) | factorization
  ---------+------+----------------------------------------
         4 |  1   | 2^2
        32 |  2   | 2^2 * 2^3
       288 |  3   | 2^2 * 2^3 * 3^2
      4608 |  4   | 2^2 * 2^3 * 3^2 * 2^4
    115200 |  5   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2
   3110400 |  6   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3
  99532800 |  7   | 2^2 * 2^3 * 3^2 * 2^4 * 5^2 * 3^3 * 2^5

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001694, A005117, A052146, A077761, A118914, A246547, A246655, A376679, A385378, A385380 (indices of records).

f[p_, e_] := Floor[(Sqrt[8*e + 9] - 3)/2]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> (sqrtint(8*x+9)-1)\2 , factor(n)[, 2]));

Additive with a(p^e) = A052146(e+1).
a(n) = 0 if and only if n is squarefree (A005117).
a(A385380(n)) = n-1.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{k>=1} P(k*(k+3)/2) = 0.49006911093767425812..., and P is the prime zeta function.

A345236 Triangle read by rows: The rightmost column contains the terms of A002262 starting at A002262(1). Each time a column's value is zero (except for a(0)), the column to its left starts at the next term in A002262, or if that column does not yet exist, it starts at A002262(2).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 2, 0, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 4, 1, 2, 5, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 3, 1, 0, 0, 4, 1, 0, 0, 5, 1, 0, 0, 6, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 3

Offset: 0

John-Vincent Saddic, Jul 14 2021

The first row with k columns is the A006893(k)-th. The last row with k columns comprises the first k terms of A006893.

			Triangle begins as:
  0
  1
  1 0
  1 1
  1 2
  1 0 0
  1 0 1
  1 0 2
  1 0 3
  1 1 0

John-Vincent Saddic, Java code to print the n-th row of the triangle

Cf. A002262, A052146.

First row with n values: A006893(n).

See Links. Rows are printed with values concatenated. Values greater than 10 are represented between parentheses, e.g., row 100 is 113(10).

To calculate the values of the n-th row:
c(m) = floor((sqrt(9 + 8*m) - 3)/2) = A052146(m+1).
r(m) = m - (c(m)^2)/2 - 3*c(m)/2 = A002262(m+1).
The last value of row m is r(m), the second to last value is r(c(m)), the third to last value is r(c(c(m))), and so on until c(m) equals 0.

cp's OEIS Frontend

A121431 Number of subpartitions of partition P=[0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,...] (A052146).

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A241719 Number T(n,k) of compositions of n into distinct parts with exactly k descents; triangle T(n,k), n>=0, 0<=k<=max(floor((sqrt(1+8*n)-3)/2),0), read by rows.

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A238406 Number T(n,k) of partitions of n into k parts such that every i-th smallest part (counted with multiplicity) is different from i; triangle T(n,k), n>=0, 0<=k<=floor((sqrt(9+8*n)-3)/2) read by rows.

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Maple

Mathematica

A113402 Algebraic degree of cos(Pi/n) for constructible n-gons (A003401).

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Mathematica

A385379 The maximum possible number of distinct composite prime powers (A246547) in the factorization of n into prime powers.

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A345236 Triangle read by rows: The rightmost column contains the terms of A002262 starting at A002262(1). Each time a column's value is zero (except for a(0)), the column to its left starts at the next term in A002262, or if that column does not yet exist, it starts at A002262(2).

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