Aldo González Lorenzo - cp's OEIS frontend

A260550 a(n) is the number of 2 X 2 matrices with entries in {1, ..., n} that are not the product of two 2 X 2 positive integer matrices.

1, 15, 75, 237, 559, 1157, 2055, 3471, 5449, 8131, 11633, 16361, 22041, 29349, 38329, 48839, 61325, 76479, 93957, 114717, 138041, 164153, 194505, 229625, 268259, 311031, 359719, 413245, 472145, 537835, 608837, 688121, 774877, 867549, 971403, 1080637, 1198233, 1326059, 1467029, 1617451, 1777881, 1948219, 2132381, 2329081, 2539351

Offset: 1

Aldo González Lorenzo, Jul 29 2015

a(n) <= A000583(n), which is the number of 2 X 2 matrices with entries in {1, ..., n}.
a(n) >= A005917(n), which is the number of 2 X 2 matrices with entries in {1, ..., n} that contain the element 1. All such matrices are not decomposable as a product of 2 X 2 positive integer matrices.
This definition is a generalization of the notion of prime numbers to the family of 2 X 2 positive integer matrices. Since the matrices do not contain 0, max(A*B) > max(A) and max(A*B) > max(B). Thus, for every matrix there is a finite number of possible decompositions to check.

			The matrix [2,2;3,3] is decomposable: [2,2;3,3] = [1,1;1,2] * [1,1;1,1]. However, the matrix [2,3;3;2] is not decomposable.

Michael S. Branicky, Table of n, a(n) for n = 1..60
Michael S. Branicky, Python program
Aldo González Lorenzo, Scilab function for computing this sequence
P. F. Rivett and N. I. P. Mackinnon, Prime Matrices, The Mathematical Gazette, Vol. 70, No. 454 (Dec., 1986), pp. 257-259.

Cf. A000583, A005917.

Python
```
# See Branicky link.
```

A197168 Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 2.

Original entry on oeis.org

0, 0, 2, 6, 18, 30, 50, 70, 98, 118, 162, 198, 226, 270, 338, 366, 426, 486, 554, 622, 682, 726, 842, 926, 986, 1062, 1202, 1270, 1362, 1470, 1562, 1678, 1802, 1878, 2066, 2158, 2250, 2390, 2602, 2694, 2818, 2974, 3114, 3278, 3434, 3526, 3786, 3966, 4090, 4254, 4490

Offset: 0

Aldo González Lorenzo, Oct 10 2011

nonn

It is also the number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is -2.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Table[cnt = 0; Do[If[a*d-b*c == 2, cnt++], {a, n}, {b, n}, {c, n}, {d, n}]; cnt, {n, 50}] (* T. D. Noe, Oct 11 2011 *)

a(n) = A209973(n) - 4*n - 2 for n >= 2. - Chai Wah Wu, Nov 28 2016

a(0) added by Chai Wah Wu, Nov 27 2016

A196227 Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 1.

Original entry on oeis.org

0, 0, 2, 8, 14, 28, 34, 56, 70, 92, 106, 144, 158, 204, 226, 256, 286, 348, 370, 440, 470, 516, 554, 640, 670, 748, 794, 864, 910, 1020, 1050, 1168, 1230, 1308, 1370, 1464, 1510, 1652, 1722, 1816, 1878, 2036, 2082, 2248, 2326, 2420, 2506, 2688, 2750, 2916, 2994

Offset: 0

Aldo González Lorenzo, Sep 29 2011

nonn

It is also the number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is -1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A171503 (determinants of matrices that include zero), A209978, A210000.

a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)-2 + 4*numtheory[phi](n))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, May 05 2020

Table[cnt = 0; Do[If[a*d-b*c == 1, cnt++], {a, n}, {b, n}, {c, n}, {d, n}]; cnt, {n, 50}] (* T. D. Noe, Oct 11 2011 *)

a(n) = if(n < 1, 0, 4*sum(k=1, n, eulerphi(k)) - 2*(n + 1)) \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = A171503(n) - (2*n + 1) for n > 0.
a(n) = -2*(n + 1) + 4*Sum_{k=1..n} phi(k) for n > 0.
a(n) = 2 * A209978(n). (End)

a(0)=0 prependend by Andrew Howroyd, May 05 2020

A190481 Number of distinct integers with n digits which are the image of integers by the function Reverse and Add!.

Original entry on oeis.org

4, 14, 93, 256, 1793, 4872, 34107, 92590, 648154, 1759313, 12315269, 33427272, 233991155, 635119194, 4445835138, 12067267861, 84470877438, 229278099157, 1604946701532, 4356283914175, 30493987422124, 82769394462323, 579385761306789, 1572618495070552

Offset: 1

Aldo González Lorenzo, May 25 2011

a(n) is the cardinality of the set of Image(Reverse and Add!) intersected with [10^(n-1), 10^n[. Here we suppose that the domain of the function Reverse and Add! is {1, 2, 3, ...}

There are 4, 50, 450, 4590, 45405,... (A232731) ways to obtain integers with n = 1,2,... digits as images under the function "Reverse and add!", but many result in the same image and are counted here only once. Example: 11+digrev(11) = 22 and 20+digrev(20)=22 contribute only once to the set of distinct images at n=2. - R. J. Mathar, Jun 17 2011

			Example: let RaA(x) be the function Reverse and Add!, then:
RaA(1)=2
RaA(2)=4
RaA(3)=6
RaA(4)=8
RaA(5)=10
RaA(6)=11, ...
So a(1) is the cardinal of {2,4,6,8}, which is 4:

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100

Cf. A056964, A067030.

A055642 := proc(n) max(1,1+ilog10(n)) ; end proc:
A056964 := proc(n) n+digrev(n) ; end proc:
A190481 := proc(n) local s,i,ra ; s := {} ; for i from 1 to 10^n do ra := A056964(i) ; if A055642( ra) = n then s := s union {ra}  ; end if; end do: nops(s) ; end proc:
for n from 1 do print(n,A190481(n)) ; end do: # R. J. Mathar, Jun 17 2011

Empirical g.f.: x*(4 + 18*x + 23*x^2 - 29*x^3 - 58*x^4 - 34*x^5 - 81*x^6 - 45*x^7 - 32*x^8 - 9*x^9) / ((1 + x)*(1 - 19*x^2)*(1 - 2*x + x^2 - x^3)*(1 + 2*x + x^2 + x^3)). - Colin Barker, Mar 20 2017

a(9)-a(10) from Lars Blomberg, Dec 01 2013

a(11)-a(24) from Hiroaki Yamanouchi, Sep 04 2014

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User: Aldo González Lorenzo

A260550 a(n) is the number of 2 X 2 matrices with entries in {1, ..., n} that are not the product of two 2 X 2 positive integer matrices.

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A197168 Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 2.

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A196227 Number of 2 X 2 integer matrices with elements from {1,...,n} whose determinant is 1.

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Maple

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A190481 Number of distinct integers with n digits which are the image of integers by the function Reverse and Add!.

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Author

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Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions