A140436 -id:A140436 - cp's OEIS frontend

A034891 Number of different products of partitions of n; number of partitions of n into prime parts (1 included); number of distinct orders of Abelian subgroups of symmetric group S_n.

1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 81, 98, 117, 140, 166, 196, 231, 271, 317, 369, 429, 496, 573, 660, 758, 869, 993, 1133, 1290, 1465, 1662, 1881, 2125, 2397, 2699, 3035, 3407, 3820, 4276, 4780, 5337, 5951, 6628, 7372, 8191, 9090

Offset: 0

Wouter Meeussen

a(n) = length of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012

Number of partitions of n into noncomposite parts. - Omar E. Pol, Jun 23 2022

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from T. D. Noe)
Index entries for sequences related to partitions

Cf. A000041, A000792, A000793, A009490, A008578, A140436, A353188.

a034891 = length . a212721_row  -- Reinhard Zumkeller, Jun 14 2012

b:= proc(n, i) option remember; `if`(n=0, 1, (p->
      `if`(i<0, 0, b(n, i-1)+ `if`(p>n, 0,
         b(n-p, i))))(`if`(i<1, 1, ithprime(i))))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 15 2013

Table[ Length[ Union[ Apply[ Times, Partitions[ n], 1]]], {n, 30}]
CoefficientList[ Series[ (1/(1 - x)) Product[1/(1 - x^Prime[i]), {i, 100}], {x, 0, 50}], x] (* Robert G. Wilson v, Aug 17 2013 *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n == 0, 1, If[i<0, 0, b[n, i-1] + If[p>n, 0, b[n-p, i]]]]]; a[n_] := b[n, PrimePi[n] ]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 05 2015, after Alois P. Heinz *)

[Partitions(n, parts_in=(prime_range(n+1)+[1])).cardinality() for n in xsrange(1000)] # Giuseppe Coppoletta, Jul 11 2016

G.f.: (1/(1-x))*(1/Product_{k>0} (1-x^prime(k))). a(n) = (1/n)*Sum_{k=1..n} A074372(k)*a(n-k). Partial sums of A000607. - Vladeta Jovovic, Sep 19 2002

a(n) = A000041(n) - A353188(n). - Omar E. Pol, Jun 23 2022

More terms from Vladeta Jovovic

a(0)=1 from Michael Somos, Feb 05 2011

A190427 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,2,1) and []=floor.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n) = [(b*n+c)*r] - b*[n*r] - [c*r].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427 - A190430
(golden ratio,3,0):  A140397 - A190400
(golden ratio,3,1):  A140431 - A190435
(golden ratio,3,2):  A140436 - A190439

			a(1)=[3r]-2[r]-1=4-3-1=1.
a(2)=[5r]-2[2r]-1=8-6-1=1.
a(3)=[7r]-2[3r]-1=11-8-1=2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001622, A078588, A190428, A190429, A190430.

[Floor((2*n+1)*(1+Sqrt(5))/2) - 2*Floor(n*(1+Sqrt(5))/2) - 1: n in [1..100]]; // G. C. Greubel, Apr 06 2018

r = GoldenRatio; b = 2; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}] (* A190427 *)
Flatten[Position[t, 0]] (* A190428 *)
Flatten[Position[t, 1]] (* A190429 *)
Flatten[Position[t, 2]] (* A190430 *)
Table[Floor[(2*n+1)*GoldenRatio] - 2*Floor[n*GoldenRatio] -1, {n,1,100}] (* G. C. Greubel, Apr 06 2018 *)

for(n=1,100, print1(floor((2*n+1)*(1+sqrt(5))/2) - 2*floor(n*(1+sqrt(5))/2) - 1, ", ")) \\ G. C. Greubel, Apr 06 2018

from mpmath import mp, phi
from sympy import floor
mp.dps=100
def a(n): return floor((2*n + 1)*phi) - 2*floor(n*phi) - 1
print([a(n) for n in range(1, 132)]) # Indranil Ghosh, Jul 02 2017

a(n) = [(2*n+1)*r] - 2*[n*r] - 1, where r=(1+sqrt(5))/2.

A190440 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,0) and []=floor.

Original entry on oeis.org

2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439

Cf. A190889, A190442, A190443, A190251.

r = GoldenRatio;
f[n_] := Floor[4*n*r] - 4*Floor[n*r];
t = Table[f[n], {n, 1, 320}] (* A190440 *)
Flatten[Position[t, 0]]  (* A190240 *)
Flatten[Position[t, 1]]  (* A190249 *)
Flatten[Position[t, 2]]  (* A190442 *)
Flatten[Position[t, 3]]  (* A190443 *)
Flatten[Position[t, 4]]  (* A190248 *)

a(n)=[4nr]-4[nr], where r=golden ratio.

A190431 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.

Original entry on oeis.org

2, 1, 3, 2, 0, 2, 1, 3, 2, 1, 3, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 1, 3, 1, 0, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 1, 3, 1, 0, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 3, 2, 0, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 1, 3, 1, 0, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 1, 3, 1, 0, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 3, 1, 0, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 3, 2, 1, 3, 1, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n) = [(b*n+c)*r] - b*[n*r] - [c*r].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427 - A190430
(golden ratio,3,0):  A140397 - A190400
(golden ratio,3,1):  A140431 - A190435
(golden ratio,3,2):  A140436 - A190439

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190432, A190433, A190434, A190435.

[Floor((3*n+1)*(1+Sqrt(5))/2) - 3*Floor(n*(1+Sqrt(5))/2) - 1: n in [1..100]]; // G. C. Greubel, Apr 06 2018

r = GoldenRatio; b = 3; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}] (* A190431 *)
Flatten[Position[t, 0]] (* A190432 *)
Flatten[Position[t, 1]] (* A190433 *)
Flatten[Position[t, 2]] (* A190434 *)
Flatten[Position[t, 3]] (* A190435 *)

for(n=1,100, print1(floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1, ", ")) \\ G. C. Greubel, Apr 06 2018

a(n) = floor((3*n+1)*(1+sqrt(5))/2) - 3*floor(n*(1+sqrt(5))/2) - 1. - G. C. Greubel, Apr 06 2018

A190436 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

Original entry on oeis.org

2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 0, 2, 0, 2, 1, 0, 2, 1, 3, 1, 0, 2, 1, 0, 2, 1, 2, 1, 0, 2, 1, 3, 2, 0, 2, 1, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427 - A190430
(golden ratio,3,0):  A140397 - A190400
(golden ratio,3,1):  A140431 - A190435
(golden ratio,3,2):  A140436 - A190439
(golden ratio,4,c):  A140440 - A190461

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A190437, A190438, A190439, A190440.

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)

A190445 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,1) and []=floor.

Original entry on oeis.org

3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439
(golden ratio,4,c):  A190440-A190461

Cf. A190446-A190450, A190427.

r = GoldenRatio; b = 4; c = 1;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]

A190457 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,3) and []=floor.

Original entry on oeis.org

3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 4, 2, 1, 3, 2, 4, 3, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 4, 3

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439
(golden ratio,4,c):  A190440-A190461

Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.

Cf. A190458-A190461 and A190463.

r = GoldenRatio; b = 4; c = 3;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]

A302253 Positions of 3 in A190436.

Original entry on oeis.org

8, 21, 29, 42, 55, 63, 76, 97, 110, 118, 131, 144, 152, 165, 186, 199, 207, 220, 241, 254, 262, 275, 288, 296, 309, 330, 343, 351, 364, 377, 385, 398, 406, 419, 432, 440, 453, 474, 487, 495, 508, 521, 529, 542, 563, 576, 584, 597, 618, 631, 639, 652, 665, 673, 686, 707, 720, 728

Offset: 1

G. C. Greubel, Apr 04 2018

nonn

Write a(n) = [(bn+c)r] - b[nr] - [cr]. If r>0 and b and c are integers satisfying b >= 2 and 0 <= c <= b-1, then 0 <= a(n) <= b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439
(golden ratio,4,c):  A140440-A190461

G. C. Greubel, Table of n, a(n) for n = 1..20000

Cf. A190436, A190437, A190438, A190439.

r = GoldenRatio; b = 3; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 500}] (* A190436 *)
Flatten[Position[t, 0]] (* A190437 *)
Flatten[Position[t, 1]] (* A190438 *)
Flatten[Position[t, 2]] (* A190439 *)
Flatten[Position[t, 3]] (* A302253 *)

A190451 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,2) and []=floor.

Original entry on oeis.org

2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 2, 1, 3, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 0, 3, 1, 0, 2, 1, 3, 2, 0, 3, 1, 4, 2, 1, 3, 2, 0, 3, 1, 3, 2, 0, 3, 1, 0

Offset: 1

Clark Kimberling, May 10 2011

nonn

Write a(n)=[(bn+c)r]-b[nr]-[cr].  If r>0 and b and c are integers satisfying b>=2 and 0<=c<=b-1, then 0<=a(n)<=b.  The positions of 0 in the sequence a are of interest, as are the position sequences for 1,2,...,b.  These b+1 position sequences comprise a partition of the positive integers.
Examples:
(golden ratio,2,0):  A078588, A005653, A005652
(golden ratio,2,1):  A190427-A190430
(golden ratio,3,0):  A140397-A190400
(golden ratio,3,1):  A140431-A190435
(golden ratio,3,2):  A140436-A190439

Cf. A190428, A190453-A190455.

r = GoldenRatio; b = 4; c = 2;
f[n_] := Floor[(b*n + c)*r] - b*Floor[n*r] - Floor[c*r];
t = Table[f[n], {n, 1, 320}]
Flatten[Position[t, 0]]
Flatten[Position[t, 1]]
Flatten[Position[t, 2]]
Flatten[Position[t, 3]]
Flatten[Position[t, 4]]

cp's OEIS Frontend

A034891 Number of different products of partitions of n; number of partitions of n into prime parts (1 included); number of distinct orders of Abelian subgroups of symmetric group S_n.

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A190427 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,2,1) and []=floor.

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A190440 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,0) and []=floor.

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A190431 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.

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A190436 a(n) = [(b*n+c)*r] - b*[n*r] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.

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A190445 [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,1) and []=floor.

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A190457 a(n) = [(bn+c)r]-b[nr]-[cr], where (r,b,c)=(golden ratio,4,3) and []=floor.

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A302253 Positions of 3 in A190436.

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A190427 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,2,1) and []=floor.

A190431 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,1) and []=floor.

A190436 a(n) = [(bn+c)r] - b[nr] - [c*r], where (r,b,c)=(golden ratio,3,2) and []=floor.