A081341 -id:A081341

A134309

Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

+0
14

1, 0, 1, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

As infinite lower triangular matrices, binomial transform of A134309 = A082137. A134309 * A007318 = A055372. A134309 * [1,2,3,...] = A057711: (1, 2, 6, 16, 40, 96, 224,...).

Triangle read by rows given by [0,0,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

LINKS

Table of n, a(n) for n=0..97.

FORMULA

Triangle, T(0,0) = 1, then for n > 0, n zeros followed by 2^(n-1). Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.

G.f.: (1 - y*x)/(1 - 2*y*x). - Philippe Deléham, Feb 04 2012

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Feb 04 2012

Diagonal is A011782, other elements are 0. - M. F. Hasler, Mar 29 2022

EXAMPLE

Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

1;

0, 1;

0, 0, 2;

0, 0, 0, 4;

0, 0, 0, 0, 8;

0, 0, 0, 0, 0, 16;

...

MATHEMATICA

Join[{1}, Flatten[Table[Join[{PadRight[{}, n], 2^(n-1)}], {n, 20}]]] (* Harvey P. Dale, Jan 04 2024 *)

PROG

(PARI) A134309(r, c)=if(r==c, 2^max(r-1, 0), 0) \\ M. F. Hasler, Mar 29 2022

CROSSREFS

Cf. A011782 (diagonal elements: 1 followed by 1, 2, 4, 8, ... = A000079: 2^n).

Cf. A055372, A057711, A082137.

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Oct 19 2007

STATUS

approved

A192396

Square array T(n, k) = floor(((k+1)^n - (1+(-1)^k)/2)/2) read by antidiagonals.

+0
2

0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 4, 4, 2, 0, 0, 8, 13, 8, 2, 0, 0, 16, 40, 32, 12, 3, 0, 0, 32, 121, 128, 62, 18, 3, 0, 0, 64, 364, 512, 312, 108, 24, 4, 0, 0, 128, 1093, 2048, 1562, 648, 171, 32, 4, 0, 0, 256, 3280, 8192, 7812, 3888, 1200, 256, 40, 5, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

T(n,k) is the number of compositions of odd natural numbers into n parts <=k.

LINKS

G. C. Greubel, Antidiagonals n = 0..50, flattened

Adi Dani, Restricted compositions of natural numbers

EXAMPLE

T(2,4)=12: there are 12 compositions of odd natural numbers into 2 parts <=4

1: (0,1), (1,0);

3: (1,2), (2,1), (0,3), (3,0);

5: (1,4), (4,1), (2,3), (3,2);

7: (3,4), (4,3).

The table starts

0, 0, 0, 0, 0, 0, ... A000004;

0, 1, 1, 2, 2, 3, ... A004526;

0, 2, 4, 8, 12, 18, ... A007590;

0, 4, 13, 32, 62, 108, ... A036487;

0, 8, 40, 128, 312, 648, ... A191903;

0, 16, 121, 512, 1562, 3888, ... A191902;

. . . . ...

with columns: A000004, A000079, A003462, A004171, A128531, A081341, ... .

Antidiagonal triangle begins:

0;

0, 0;

0, 1, 0;

0, 2, 1, 0;

0, 4, 4, 2, 0;

0, 8, 13, 8, 2, 0;

0, 16, 40, 32, 12, 3, 0;

0, 32, 121, 128, 62, 18, 3, 0;

0, 64, 364, 512, 312, 108, 24, 4, 0;

MAPLE

A192396 := proc(n, k) (k+1)^n-(1+(-1)^k)/2 ; floor(%/2) ; end proc:

seq(seq( A192396(d-k, k), k=0..d), d=0..10) ; # R. J. Mathar, Jun 30 2011

MATHEMATICA

T[n_, k_]:= Floor[((k+1)^n - (1+(-1)^k)/2)/2];

Table[T[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten

PROG

(Magma)

A192396:= func< n, k | Floor(((k+1)^n - (1+(-1)^k)/2)/2) >;

[A192396(n-k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 11 2023

(SageMath)

def A192396(n, k): return ((k+1)^n - ((k+1)%2))//2

flatten([[A192396(n-k, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 11 2023

CROSSREFS

Rows: A000004, A004526, A007590, A036487, A191902, A191902.

Columns: A000004, A000079, A003462, A004171, A081341, A128531.

KEYWORD

nonn,tabl,easy

AUTHOR

Adi Dani, Jun 29 2011

STATUS

approved

A034494

a(n) = (7^n+1)/2.

+0
12

1, 4, 25, 172, 1201, 8404, 58825, 411772, 2882401, 20176804, 141237625, 988663372, 6920643601, 48444505204, 339111536425, 2373780754972, 16616465284801, 116315256993604, 814206798955225, 5699447592686572

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Binomial transform of A081341. Inverse binomial transform of A081342. [R. J. Mathar, Oct 23 2008]

Number of compositions of even natural numbers into n parts <=6. [Adi Dani, May 28 2011]

From Charlie Marion, Jun 24 2011: (Start)

a(n)+(a(n)+1)+...+(a(n+1)-7^n-1)=(a(n+1)-7^n)+...+(a(n+1)-1). Let

S(2n) and S(2n+1) be the sets of addends on the left- and right-hand

sides, respectively, of the preceding equations. Then, since the

intersection of any 2 different S(i) is null and the union of all of

them is the positive integers, {S(i)} forms a partition of the

positive integers. See also A034659.

In general, for k>0, let b(n)=((4k+3)^n+1)/2. Then b(n)+(b(n)+1)+ ...

+(b(n+1)-(4k+3)^n-1)=k*((b(n+1)-(4k+3)^n)+ ... +(b(n+1)-1)). Then,

for each k, the set of addends on the two sides of these equations

also forms a partition of the positive integers. Also, with b(0)=1,

b(n)=(4k+3)*b(n-1)-(2k+1).

For k>0, let c(0)=1 and, for n>0, c(n)=(2*(2k+1))^n/2. Then the

sequence b(0),b(1),... is the binomial transform of the sequence

c(0),c(1),....

For k>0, let d(2n)=(2k+1)^(2n) and d(2n+1)=0. Then the sequence

b(0),b(1),... is the (2k+2)nd binomial transform of the sequence

d(0),d(1),.... (End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (8, -7).

FORMULA

E.g.f.: exp(4*x)*cosh(3*x). - Paul Barry, Apr 20 2003

a(n) = 7a(n-1) - 3, a(0) = 1.

G.f.: (1-4*x)/((1-x)*(1-7*x)). - Philippe Deléham, Jul 11 2005

a(n) = 8*a(n-1)-7*a(n-2), a(0)=1, a(1)=4. [Philippe Deléham, Nov 15 2008]

a(n) = ((4+sqrt(9))^n+(4-sqrt(9))^n)/2. [Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008]

EXAMPLE

From Adi Dani, May 28 2011: (Start)

a(2)=25: there are 25 compositions of even numbers into 2 parts <=6:

(0,0)

(0,2),(2,0),(1,1)

(0,4),(4,0),(1,3),(3,1),(2,2)

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

(2,6),(6,2),(3,5),(5,3),(4,4)

(4,6),(6,4),(5,5)

(6,6)

(end)

MAPLE

A034494:=n->(7^n+1)/2: seq(A034494(n), n=0..30); # Wesley Ivan Hurt, Apr 09 2017

PROG

(Magma) [(7^n+1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 16 2011

(PARI) a(n)=(7^n+1)/2 \\ Charles R Greathouse IV, Jul 02 2013

(PARI) Vec((1-4*x)/((1-x)*(1-7*x)) + O(x^100)) \\ Altug Alkan, Nov 01 2015

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A169604

a(n) = 3*6^n.

+0
6

3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

a(n) = A081341(n+1).

Essentially first differences of A125682.

Binomial transform of A005053 without initial term 1.

Second binomial transform of A164346.

Inverse binomial transform of A169634.

Second inverse binomial transform of A103333 without initial term 1.

Contribution from Reinhard Zumkeller, May 02 2010: (Start)

a(n) = 3*A000400(n) = A000400(n+1)/2;

subsequence of A003586; a(n)=A003586(A014105(n)) for n<6. (End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Index entries for linear recurrences with constant coefficients, signature (6).

FORMULA

a(n) = 6*a(n-1) for n > 0; a(0) = 3.

G.f.: 3/(1-6*x).

PROG

(Magma) [ 3*6^n: n in [0..19] ];

(PARI) a(n)=3*6^n \\ Charles R Greathouse IV, Oct 16 2015

CROSSREFS

Cf. A081341, A125682 ((6^n-1)*3/5), A005053 (expand (1-2x)/(1-5x)), A164346 (3*4^n), A169634 (3*7^n), A103333 (expand (1-5x)/(1-8x)).

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 04 2010

STATUS

approved

A261399

a(1)=1; thereafter a(n) = (2/5)*(9*6^(n-2)+1).

+0
2

1, 4, 22, 130, 778, 4666, 27994, 167962, 1007770, 6046618, 36279706, 217678234, 1306069402, 7836416410, 47018498458, 282110990746, 1692665944474, 10155995666842, 60935974001050, 365615844006298, 2193695064037786, 13162170384226714, 78973022305360282, 473838133832161690

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Partial sums of A081341. - Klaus Purath, Jul 28 2020

LINKS

Table of n, a(n) for n=1..24.

K. Hong, H. Lee, H. J. Lee and S. Oh, Small knot mosaics and partition matrices, J. Phys. A: Math. Theor. 47 (2014) 435201; arXiv:1312.4009 [math.GT], 2013-2014. See Cor. 2.

Index entries for linear recurrences with constant coefficients, signature (7,-6).

Index entries for sequences related to knots

FORMULA

G.f.: x-2*x^2*(-2+3*x) / ( (6*x-1)*(x-1) ). - R. J. Mathar, Aug 19 2015

a(n) = 2*A199412(n-2), n>1. - R. J. Mathar, Aug 19 2015

From Klaus Purath, Jul 28 2020: (Start)

a(n) = 7*a(n-1) - 6*a(n-2), n > 2.

a(n) = 6*a(n-1) - 2, n > 1.

a(n) = 3*6^(n-2) + a(n-1), n > 1.

(End)

CROSSREFS

The number 22, the third term here, is the same 22 seen in A261400 and illustrated in a link in that entry.

Cf. A199412.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Aug 19 2015

STATUS

approved

A055372

Invert transform of Pascal's triangle A007318.

+0
14

1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 32, 48, 32, 8, 16, 80, 160, 160, 80, 16, 32, 192, 480, 640, 480, 192, 32, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 128, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 256, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

T(n,k) is the number of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence. For example, T(2,1)=4 because we have {(01)},{(10)},{(0),(1)},{(1),(0)}. - Geoffrey Critzer, Apr 06 2013

LINKS

Table of n, a(n) for n=0..54.

N. J. A. Sloane, Transforms

Index entries for triangles and arrays related to Pascal's triangle

FORMULA

a(n,k) = 2^(n-1)*C(n, k), for n>0.

G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).

As an infinite lower triangular matrix, equals A134309 * A007318. - Gary W. Adamson, Oct 19 2007

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Feb 05 2012

EXAMPLE

Triangle begins:

1;

1, 1;

2, 4, 2;

4, 12, 12, 4;

8, 32, 48, 32, 8;

...

MATHEMATICA

nn=10; f[list_]:=Select[list, #>0&]; a=(x+y x)/(1-(x+y x)); Map[f, CoefficientList[Series[1/(1-a), {x, 0, nn}], {x, y}]]//Grid (* Geoffrey Critzer, Apr 06 2013 *)

CROSSREFS

Row sums give A081294. Cf. A000079, A007318, A055373, A055374.

Cf. A134309.

T(2n,n) gives A098402.

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, May 16 2000

STATUS

approved

A081340

(5^n+(-1)^n)/2.

+0
5

1, 2, 13, 62, 313, 1562, 7813, 39062, 195313, 976562, 4882813, 24414062, 122070313, 610351562, 3051757813, 15258789062, 76293945313, 381469726562, 1907348632813, 9536743164062, 47683715820313, 238418579101562

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Binomial transform of A003665. 2nd binomial transform of (1,0,9,0,81,0,729,0,..). Case k=2 of family of recurrences a(n)=2k*a(n-1)-(k^2-9)*a(n-2), a(0)=0, a(1)=k. A003665 is case k=1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (4,5).

FORMULA

a(n) = 4*a(n-1) + 5*a(n-2), a(0)=1, a(1)=2.

G.f.: (1-2*x)/((1+x)*(1-5*x)).

E.g.f.: exp(2*x) * cosh(3*x).

a(n) = ((2+sqrt(9))^n+(2-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008

a(n) = sum( k=0..n, A201730(n,k)*8^k ). - Philippe Deléham, Dec 06 2011

MATHEMATICA

CoefficientList[Series[(1 - 2 x) / ((1 + x) (1 - 5 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 08 2013 *)

PROG

(Sage) [lucas_number2(n, 4, -5)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

(PARI) a(n)=(5^n+(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. A081341, A087404.

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Mar 18 2003

STATUS

approved

A191348

Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

+0
3

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 2, 1, 0, 8, 20, 7, 2, 1, 0, 16, 68, 26, 8, 3, 1, 0, 32, 232, 97, 32, 14, 3, 1, 0, 64, 792, 362, 128, 72, 15, 3, 1, 0, 128, 2704, 1351, 512, 376, 81, 16, 3, 1, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..55.

FORMULA

For each row n >= 0 let T(n,0)=1 and T(n,1) = ceiling(sqrt(n)), then for each column k >= 2: T(n,k) = T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 23 2019

EXAMPLE

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

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

1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, ...

1, 2, 7, 26, 97, 362, 1351, 5042, 18817, 70226, 262087, ...

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, ...

1, 3, 14, 72, 376, 1968, 10304, 53952, 282496, 1479168, 7745024, ...

1, 3, 15, 81, 441, 2403, 13095, 71361, 388881, 2119203, 11548575, ...

1, 3, 16, 90, 508, 2868, 16192, 91416, 516112, 2913840, 16450816, ...

1, 3, 17, 99, 577, 3363, 19601, 114243, 665857, 3880899, 22619537, ...

1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, ...

1, 4, 26, 184, 1316, 9424, 67496, 483424, 3462416, 24798784, 177615776, ...

1, 4, 27, 196, 1433, 10484, 76707, 561236, 4106353, 30044644, 219825387, ...

1, 4, 28, 208, 1552, 11584, 86464, 645376, 4817152, 35955712, 268377088, ...

1, 4, 29, 220, 1673, 12724, 96773, 736012, 5597777, 42574180, 323800109, ...

1, 4, 30, 232, 1796, 13904, 107640, 833312, 6451216, 49943104, 386642400, ...

...

PROG

(PARI) T(n, k) = if (k==0, 1, if (k==1, ceil(sqrt(n)), T(n, k-2)*(n-T(n, 1)^2) + T(n, k-1)*T(n, 1)*2));

matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 23 2019

CROSSREFS

Row 1 is A000007, row 2 is A011782, row 3 is A006012, row 4 is A001075, row 5 is A081294, row 6 is A098648, row 7 is A084120, row 8 is A146963, row 9 is A001541, row 10 is A081341, row 11 is A084134, row 13 is A090965.

Row 3*2 is A056236, row 4*2 is A003500, row 5*2 is A155543, row 9*2 is A003499.

Cf. A191347 which uses floor() in place of ceiling().

KEYWORD

nonn,tabl

AUTHOR

Charles L. Hohn, May 31 2011

STATUS

approved

A191347

Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

+0
3

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..65.

FORMULA

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019

T(n, k) = Sum_(i=0, floor((k+1)/2), binomial(k, 2*i)*floor(sqrtint(n))^(k-2*i)*n^i)) for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

EXAMPLE

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

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

1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, ...

1, 1, 4, 10, 28, 76, 208, 568, 1552, 4240, 11584, ...

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, ...

1, 2, 9, 38, 161, 682, 2889, 12238, 51841, 219602, 930249, ...

1, 2, 10, 44, 196, 872, 3880, 17264, 76816, 341792, 1520800, ...

1, 2, 11, 50, 233, 1082, 5027, 23354, 108497, 504050, 2341691, ...

1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, ...

1, 3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, ...

1, 3, 19, 117, 721, 4443, 27379, 168717, 1039681, 6406803, 39480499, ...

1, 3, 20, 126, 796, 5028, 31760, 200616, 1267216, 8004528, 50561600, ...

1, 3, 21, 135, 873, 5643, 36477, 235791, 1524177, 9852435, 63687141, ...

1, 3, 22, 144, 952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...

1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...

...

PROG

(PARI) T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));

matrix(9, 9, n, k, T(n-1, k-1)) \\ Michel Marcus, Aug 22 2019

(PARI) T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n, k-2)*(n-T(n, 1)^2) + T(n, k-1)*T(n, 1)*2));

matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

CROSSREFS

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.

Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.

Cf. A191348 which uses ceiling() in place of floor().

KEYWORD

nonn,tabl

AUTHOR

Charles L. Hohn, May 31 2011

STATUS

approved

A166469

Number of divisors of n which are not multiples of consecutive primes.

+0
12

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 3, 5, 2, 4, 2, 6, 4, 4, 2, 5, 3, 4, 4, 6, 2, 5, 2, 6, 4, 4, 3, 5, 2, 4, 4, 8, 2, 6, 2, 6, 4, 4, 2, 6, 3, 6, 4, 6, 2, 5, 4, 8, 4, 4, 2, 7, 2, 4, 6, 7, 4, 6, 2, 6, 4, 6, 2, 6, 2, 4, 4, 6, 3, 6, 2, 10, 5, 4, 2, 8, 4, 4, 4, 8, 2, 6, 4, 6, 4, 4, 4, 7, 2, 6, 6, 9, 2, 6, 2, 8, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Links various subsequences of A025487 with an unusual number of important sequences, including the Fibonacci, Lucas, and other generalized Fibonacci sequences (see cross-references).

If a number is a product of any number of consecutive primes, the number of its divisors which are not multiples of n consecutive primes is always a Fibonacci n-step number. See also A073485, A167447.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

FORMULA

a) If n has no prime gaps in its factorization (cf. A073491), then, if the canonical factorization of n into prime powers is the product of p_i^(e_i), a(n) is the sum of all products of one or more nonadjacent exponents, plus 1. For example, if A001221(n) = 3, a(n) = e_1*e_3 + e_1 + e_2 + e_3 + 1. If A001221(n) = k, the total number of terms always equals A000045(k+2).

The answer can also be computed in k steps, by finding the answers for the products of the first i powers, for i = 1 to i = k. Let the result of the i-th step be called r(i). r(1) = e_1 + 1; r(2) = e_1 + e_2 +1; for i > 2, r(i) = r(i-1) + e_i * r(i-2).

b) If n has prime gaps in its factorization, express it as a product of the minimum number of A073491's members possible. Then apply either of the above methods to each of those members, and multiply the results to get a(n). a(n) = A000005(n) iff n has no pair of consecutive primes as divisors.

a(n) = Sum_{d|n} (1-A296210(d)). - Antti Karttunen, Dec 15 2017

EXAMPLE

Since 3 of 30's 8 divisors (6, 15, and 30) are multiples of 2 or more consecutive primes, a(30) = 8 - 3 = 5.

MATHEMATICA

Array[DivisorSum[#, 1 &, FreeQ[Differences@ PrimePi@ FactorInteger[#][[All, 1]], 1] &] &, 105] (* Michael De Vlieger, Dec 16 2017 *)

PROG

(PARI)

A296210(n) = { if(1==n, return(0)); my(ps=factor(n)[, 1], pis=vector(length(ps), i, primepi(ps[i])), diffsminusones = vector(length(pis)-1, i, (pis[i+1]-pis[i])-1)); !factorback(diffsminusones); };

A166469(n) = sumdiv(n, d, !A296210(d)); \\ Antti Karttunen, Dec 15 2017

CROSSREFS

Cf. A000040, A006094, A104210, A296210.

A(A002110(n)) = A000045(n+2); A(A097250(n)) = A000032(n+1). For more relationships involving Fibonacci and Lucas numbers, see A166470-A166473, comment on A081341.

A(A061742(n)) = A001045(n+2); A(A006939(n)) = A000085(n+1); A(A212170(n)) = A000142(n+1). A(A066120(n)) = A166474(n+1).

KEYWORD

nonn

AUTHOR

Matthew Vandermast, Nov 05 2009

EXTENSIONS

Edited by Matthew Vandermast, May 24 2012

STATUS

approved