A125666 -id:A125666 - cp's OEIS frontend

A348072 Numbers k such that omega(k) = 8.

9699690, 11741730, 13123110, 14804790, 15825810, 16546530, 17160990, 17687670, 18888870, 19399380, 20030010, 20281170, 20930910, 21111090, 21411390, 21637770, 21951930, 23130030, 23393370, 23483460, 23993970, 24534510, 25555530, 25571910, 26193090, 26246220, 26816790, 26996970

Offset: 1

David A. Corneth, Oct 10 2021

nonn

David A. Corneth, Table of n, a(n) for n = 1..10000

Row 8 of A125666.

Cf. A001221, A048692.

PARI
```
is(n) = omega(n) == 8
```

A246655(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v)
list(lim,pr=8)=if(pr==1, return(A246655(lim))); my(v=List(),pr1=pr-1,mx=prod(i=1,pr1,prime(i))); forprime(p=prime(pr),lim\mx, my(u=list(lim\p,pr1)); for(i=1,#u,listput(v,p*u[i]))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023

A348073 Numbers k such that omega(k) = 9.

Original entry on oeis.org

223092870, 281291010, 300690390, 340510170, 358888530, 363993630, 380570190, 397687290, 406816410, 417086670, 434444010, 446185740, 455885430, 458948490, 481410930, 485555070, 497668710, 504894390, 512942430, 514083570, 531990690, 538047510, 547777230, 551861310, 562582020

Offset: 1

David A. Corneth, Oct 10 2021

nonn

			562582020 = 2^2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 29 is in the sequence as it has 9 distinct prime divisors (namely 2, 3, 5, 7, 11, 13, 17, 19 and 29).

David A. Corneth, Table of n, a(n) for n = 1..10000

Row 9 of A125666.

Cf. A001221, A046312.

PARI
```
is(n) = omega(n) == 9
```

A246655(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v)
list(lim,pr=9)=if(pr==1, return(A246655(lim))); my(v=List(),pr1=pr-1,mx=prod(i=1,pr1,prime(i))); forprime(p=prime(pr),lim\mx, my(u=list(lim\p,pr1)); for(i=1,#u,listput(v,p*u[i]))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023

A190913 Sequence A190914 evaluated at the negative index -n.

Original entry on oeis.org

5, 0, 2, 9, 2, 10, 29, 14, 50, 99, 82, 220, 365, 416, 926, 1429, 1954, 3842, 5825, 8778, 15922, 24299, 38414, 66240, 102533, 165560, 276954, 434745, 707394, 1163074, 1846069, 3008302, 4900546, 7839115, 12762378, 20694684, 33271421, 54081272, 87516358, 141133157, 229065490, 370410810, 598383689, 970090922, 1568482962

Offset: 0

Reikku Kulon, May 23 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,3,0,-1)

Cf. A190914, A000045, A000073, A001608, A000040, A005117, A125666.

LinearRecurrence[{0,1,3,0,-1},{5,0,2,9,2},50] (* Vincenzo Librandi, Feb 15 2012 *)

Vec((5-3*x^2-6*x^3)/((x^2+x-1)*(x^3-x^2-x-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 08 2011

From R. J. Mathar, Jun 05 2011: (Start)
a(n) = A190914(-n).
G.f.: ( 5-3*x^2-6*x^3 ) / ( (x^2+x-1)*(x^3-x^2-x-1) ). (End)
a(n) = A000032(n) + A073145(n). - R. J. Mathar, Jun 05 2011

A190914 Expansion of ( 5-9x^2-2x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).

Original entry on oeis.org

5, 0, 6, 3, 18, 10, 57, 42, 178, 165, 566, 616, 1821, 2236, 5914, 7963, 19362, 27982, 63813, 97394, 211458, 336633, 703786, 1157544, 2350597, 3964960, 7872702, 13541691, 26425522, 46147178, 88853297, 156994354, 299165378, 533410837, 1008343310, 1810544592, 3401446413, 6140811708, 11481472994, 20815538227

Offset: 0

Reikku Kulon, May 23 2011

The sequence ..., 14, 29, 10, 2, 9, 2, 0, [5], 0, 6, 3, 18, 10, 57, 42, ...
(the number in square brackets at index 0) equals the trace of:
  [ 0 0 0 0-1 ]
  [ 1 0 0 0 0 ]
  [ 0 1 0 0 1 ]^(+n)
  [ 0 0 1 0 3 ]
  [ 0 0 0 1 0 ]
or
  [ 0 0 0 0-1 ]
  [ 1 0 0 0 0 ]
  [ 0 1 0 0 3 ]^(-n)
  [ 0 0 1 0 1 ]
  [ 0 0 0 1 0 ]
Its characteristic polynomial is (x^2 +/- x - 1) * (x^3 -/+ x^2 -/+ x - 1); these factors are Fibonacci and tribonacci polynomials.  The ratio of negative terms approaches the golden ratio; the ratio of positive terms approaches the tribonacci constant.
Prime numbers p divide a(+p) and a(-p), as the trace of a matrix M^p (mod p) is constant.
Nonprimes c very rarely divide a(+c) and a(-c) simultaneously.  The only known dual pseudoprime in the sequence is 1.
The distribution of residues induces gaps between pseudoprimes having roughly the size of c.  For example, after 1034881 there is a gap of more than one million terms without either variety of pseudoprime.
Pseudoprimes appear limited to squared primes and squarefree numbers with three or more prime factors.  11 and 13 are more common than other factors.
Positive pseudoprimes: c | a(+c)
----------------------------------------------
1
3481. . . . 59^2
17143 . . . 7 31 79
105589. . . 11 29 331
635335. . . 5 283 449
2992191 . . 3 29 163 211
3659569 . . 1913^2
Negative pseudoprimes: c | a(-c)
----------------------------------------------
1
9 . . . . . 3^2
806 . . . . 2 13 31
1419. . . . 3 11 43
6241. . . . 79^2
6721. . . . 11 13 47
12749 . . . 11 19 61
21106 . . . 2 61 173
38714 . . . 2 13 1489
146689. . . 383^2
649621. . . 7 17 53 103
1034881 . . 41 43 587

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,1,0,-1).

Cf. A190913 (extended to negative indices), A000045, A000073, A001608, A000040, A005117, A125666.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (5-9*x^2 -2*x^3)/((1+x-x^2)*(1-x-x^2-x^3)) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{0, 3, 1, 0, -1}, {5, 0, 6, 3, 18}, 40] (* G. C. Greubel, Apr 23 2019 *)

my(x='x+O('x^40)); Vec((5-9*x^2-2*x^3)/((1+x-x^2)*(1-x-x^2-x^3))) \\ G. C. Greubel, Apr 23 2019

((5-9*x^2-2*x^3)/((1+x-x^2)*(1-x-x^2-x^3))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

a(n) = A061084(n+1) + A001644(n). - R. J. Mathar, Jun 06 2011

A295644 Rectangular array, by antidiagonals; row 1 is the ordered list of all k having at most 2 unitary divisors; for n > 1, row n is the ordered list of all k having 2^n unitary divisors.

Original entry on oeis.org

1, 2, 6, 3, 10, 30, 4, 12, 42, 210, 5, 14, 60, 330, 2310, 7, 15, 66, 390, 2730, 30030, 8, 18, 70, 420, 3570, 39270, 510510, 9, 20, 78, 462, 3990, 43890, 570570

Offset: 1

Clark Kimberling, Jun 26 2018

Every positive integer occurs exactly once, so that as a sequence, this is a permutation of the positive integers.
row 1: A000961
row 2: A007774
row 3: A033992
row 4: A033993
col 1: A231209

			Northwest corner:
     1    2    3    4    5    7    8    9   11
     6   10   12   14   15   18   20   21   22
    30   42   60   66   70   78   84   90  102
   210  330  390  420  462  510  546  570  630
  2310 2730 3570 3990 4290 4620 4830 5460 5610

Cf. A034444.

As an array, essentially the same as A125666.

z = 10000;
t = Table[2^PrimeNu[n], {n, 1, z}] ;(*  A035555 *)
r[n_] := Flatten[Position[t, 2^n]]; r[1] = Join[{1}, r[1]];
v[n_, k_] := r[n][[k]];
TableForm[Table[v[n, k], {n, 1, 5}, {k, 1, 15}]]  (* A295644 array *)
Table[v[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten  (* A295644 sequence *)

cp's OEIS Frontend

A348072 Numbers k such that omega(k) = 8.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

A348073 Numbers k such that omega(k) = 9.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

A190913 Sequence A190914 evaluated at the negative index -n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A190914 Expansion of ( 5-9*x^2-2*x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A295644 Rectangular array, by antidiagonals; row 1 is the ordered list of all k having at most 2 unitary divisors; for n > 1, row n is the ordered list of all k having 2^n unitary divisors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A190914 Expansion of ( 5-9x^2-2x^3 ) / ( (1+x-x^2)*(1-x-x^2-x^3) ).