A213268 -id:A213268 - cp's OEIS frontend

A211280 Numerator of prime(n+1) - prime(n)/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

Offset: 1

Paul Curtz, Jul 05 2012

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.

Denominators are A040000.

A211280 := proc(n)
        ithprime(n+1)-ithprime(n)/2 ;
        numer(%) ;
end proc: # R. J. Mathar, Jul 10 2012

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

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

A247004 Denominator of (n+4)/gcd(n, 4)^2, a 16-periodic sequence that associates A061037 with A106617.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 09 2014

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.

			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];

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).

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

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

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}]

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

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

A219976 Denominators of the Inverse bi-binomial transform of A164558(n)/A027642(n) read downwards antidiagonals.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Dec 02 2012

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;
0.
The first column is A027641(n)/A027642(n).

			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;
1.
a(n):
1;
2,   2;
6,   6,  6,;
1,   3,  3,  1;
30, 30, 30, 30, 30;

Cf. A213268.

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 *)

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A211280 Numerator of prime(n+1) - prime(n)/2.

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A247004 Denominator of (n+4)/gcd(n, 4)^2, a 16-periodic sequence that associates A061037 with A106617.

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A219976 Denominators of the Inverse bi-binomial transform of A164558(n)/A027642(n) read downwards antidiagonals.

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