A213268 -id:A213268

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A211280

Numerator of prime(n+1) - prime(n)/2.

+10
2

2, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273, 283, 285, 285, 303, 321, 315, 315, 321, 345, 343, 357, 351, 357, 365, 375, 379

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

OFFSET

1,1

COMMENTS

Second row of the inverse semi-binomial transform of A000040(n+1) as introduced in A213268.

The list of denominators is 1, 2, 2, ... (2 repeated), so a(n) = A210497(n) for n>1.

a(n) - prime(n) = 2*prime(n+1)-prime(n)-prime(n) are prime differences (A001223) multiplied by 2, and therefore multiples of 4.

LINKS

Table of n, a(n) for n=1..73.

FORMULA

a(n) ~ n log n. Apart from the first term, a(n) = 2*prime(n+1) - prime(n). - Charles R Greathouse IV, Jul 10 2012

a(n) = prime(n+2) - A036263(n), n>1. - R. J. Mathar, Jul 10 2012

MAPLE

A211280 := proc(n)

ithprime(n+1)-ithprime(n)/2 ;

numer(%) ;

end proc: # R. J. Mathar, Jul 10 2012

MATHEMATICA

Numerator[#[[2]]-#[[1]]/2]&/@Partition[Prime[Range[80]], 2, 1] (* Harvey P. Dale, Mar 05 2023 *)

CROSSREFS

Denominators are A040000.

KEYWORD

nonn,easy,frac

AUTHOR

Paul Curtz, Jul 05 2012

STATUS

approved

A247004

Denominator of (n+4)/gcd(n, 4)^2, a 16-periodic sequence that associates A061037 with A106617.

+10
1

4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2

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

OFFSET

0,1

COMMENTS

This sequence may also be defined as the denominators of A061037(n+3)/(n+1), or also as A060819 / A109008.

One can notice that the analog numerators [numerators of (n+4)/gcd(n, 4)^2] are A106617 left-shifted 4 places.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

FORMULA

A247004 = A060819 / A109008.

(n+4) / gcd(n, 4)^2 = A188134(n+4) / 4. - Michael Somos, Sep 12 2014

a(n) = a(n+16) = a(-n), a(2*n + 1) = 1 for all n in Z. - Michael Somos, Sep 13 2014

EXAMPLE

Fractions begin:

1/4, 5, 3/2, 7, 1/2, 9, 5/2, 11, 3/4, 13, 7/2, 15, 1, 17, 9/2, 19,

5/4, 21, 11/2, 23, 3/2, 25, 13/2, 27, 7/4, 29, 15/2, 31, 2, 33, 17/2, 35,

...

Numerators begin:

1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19,

5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2, 33, 17, 35,

...

Periodic part = [4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1];

MATHEMATICA

a[n_] := (n+4)/GCD[n, 4]^2 // Denominator; Table[a[n], {n, 0, 100}]

(* or: *)

Table[{1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4}[[Mod[n, 16, 1]]], {n, 0, 100}]

PROG

(PARI) for(n=0, 100, print1(denominator((n+4)/gcd(n, 4)^2), ", ")) \\ G. C. Greubel, Aug 05 2018

(Magma) [Denominator((n+4)/Gcd(n, 4)^2): n in [0..100]]; // G. C. Greubel, Aug 05 2018

CROSSREFS

Cf. A060819, A061037, A106617, A109008, A188134, A213268.

KEYWORD

nonn,easy

AUTHOR

Jean-François Alcover and Paul Curtz, Sep 09 2014

STATUS

approved

A219976

Denominators of the Inverse bi-binomial transform of A164558(n)/A027642(n) read downwards antidiagonals.

+10
0

1, 2, 2, 6, 6, 6, 1, 3, 3, 1, 30, 30, 30, 30, 30, 1, 15, 15, 15, 15, 1, 42, 42, 210, 210, 210, 42, 42, 1, 21, 21, 105, 105, 21, 21, 1, 30, 30, 210, 210, 210, 210, 210, 30, 30, 1, 15, 15, 105, 105, 105, 105, 15, 15, 1

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

OFFSET

0,2

COMMENTS

Starting from any sequence a(k) in the first row, we define the array T(n,k) of the inverse bi-binomial transform by T(0,k) = a(k), T(n,k) = T(n-1,k+1) -2*T(n-1,k) n>0. Hence A164558(n)/A027642(n) and successive "bi-differences":

1, 3/2, 13/6, 3, 119/30, 5, 253/42, 7, 239/30, 9;

-1/2, -5/6, -4/3, -61/30, -44/15, -167/42, -106/21, -181/30, -104/15;

1/6, 1/3, 19/30, 17/15, 397/210, 61/21 , 853/210, 77/15;

0, -1/30, -2/15, -79/210, -92/105, -367/210, -314/105;

-1/30, -1/15, -23/210, -13/105, 1/210, 53/105;

0, 1/42, 2/21, 53/210, 52/105;

1/42, 1/21, 13/210, -1/105;

0, -1/30, -2/15;

-1/30, -1/15;

The first column is A027641(n)/A027642(n).

LINKS

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

EXAMPLE

Partial array of denominators:

1, 2, 6, 1, 30, 1, 42, 1, 30, 1;

2, 6, 3, 30, 15, 42, 21, 30, 15;

6, 3, 30, 15, 210, 21, 210, 15;

1, 30, 15, 210, 105, 210, 105;

30, 15, 210, 105, 210, 105;

1, 42, 21, 210, 105;

42, 21, 210, 105;

1, 30, 15;

30, 15;

a(n):

2, 2;

6, 6, 6,;

1, 3, 3, 1;

30, 30, 30, 30, 30;

MATHEMATICA

A164558[n_] := Sum[(-1)^k*Binomial[n, k]*BernoulliB[k], {k, 0, n}] // Numerator; t[0, k_?Positive] := A164558[k] / Denominator[ BernoulliB[k]]; t[n_?Positive, k_] := t[n, k] = t[n-1, k+1] - 2*t[n-1, k]; t[0, 0] = 1; t[_, _] = 0; Flatten[ Table[t[n-k , k] // Denominator, {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Dec 04 2012 *)

CROSSREFS

Cf. A213268.

KEYWORD

nonn,tabl,less

AUTHOR

Paul Curtz, Dec 02 2012

STATUS

approved

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