A082594 -id:A082594 - cp's OEIS frontend

A007442 Inverse binomial transform of primes.

2, 1, 1, -1, 3, -9, 23, -53, 115, -237, 457, -801, 1213, -1389, 445, 3667, -15081, 41335, -95059, 195769, -370803, 652463, -1063359, 1570205, -1961755, 1560269, 1401991, -11023119, 36000427, -93408425, 214275735, -450374071, 879254493, -1599245737, 2695465017

Offset: 1

N. J. A. Sloane

a(n) is the (n-1)-st difference of the first n primes. Although the magnitude of the terms appears to grow exponentially, a plot shows that the sequence a(n)/2^n has quite a bit of structure. See A082594 for an interesting application. - T. D. Noe, May 09 2003
Graph this divided by A122803 using plot2! - Franklin T. Adams-Watters
From Robert G. Wilson v, Jan 28 2020: (Start)
a(n) is odd for all n>1.
As opposed to A331573, there are terms where abs(a(n)) >= abs(a(n+1)). (End)

			a(4) = 7 - 3*5 + 3*3 - 2 = -1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..3321 (first 1000 from Franklin T. Adams-Watters)
Vaclav Kotesovec, Plot of |a(n)|^(1/n) for n = 1..10000
T. D. Noe, Plot of A007442
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Binomial Transform

Cf. A007443, A082594, A095195, A122803, A331573.

Diff[lst_List] := Table[lst[[i + 1]] - lst[[i]], {i, Length[lst] - 1}]; n=1000; dt = Prime[Range[n]]; a = Range[n]; a[[1]] = 2; Do[dt = Diff[dt]; a[[i]] = dt[[1]], {i, 2, n}]; a
u = Table[Prime[Range[k]], {k, 1, 100}];Flatten[Table[Differences[u[[k]], k - 1], {k, 1, 100}]] (* Clark Kimberling, May 15 2015 *)
t = Array[Prime, 30]; f[x_] := Rest[x] - Most[x];
Flatten[Last /@ (NestList[f, t[[1 ;; #]], (# - 1)] & /@ Range[1, 29])] (* Horst H. Manninger, Mar 22 2021 *)

vector(50, n, sum(k=0, n-1,(-1)^(n-k-1)*binomial(n-1, k)*prime(k+1))) \\ Altug Alkan, Oct 17 2015

a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) binomial(n-1, k) prime(k+1).
a(n) = A095195(n,n-1). - Alois P. Heinz, Sep 25 2013
G.f.: Sum_{k>=1} prime(k)*x^k/(1 + x)^k. - Ilya Gutkovskiy, Apr 23 2019

Incorrect conjecture concerning the sign of even terms removed by Glen Whitney, Nov 10 2024

A140119 Extrapolation for (n + 1)-st prime made by fitting least-degree polynomial to first n primes.

Original entry on oeis.org

2, 4, 8, 8, 22, -6, 72, -92, 266, -426, 838, -1172, 1432, -398, -3614, 15140, -41274, 95126, -195698, 370876, -652384, 1063442, -1570116, 1961852, -1560168, -1401888, 11023226, -36000318, 93408538, -214275608, 450374202, -879254356, 1599245876, -2695464868, 4138070460, -5539280974

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), May 08 2008

sign

Construct the least-degree polynomial p(x) which fits the first n primes (p has degree n-1 or less). Then predict the next prime by evaluating p(n+1).
Can anything be said about the pattern of positive and negative values?
Row sums of triangle A095195. - Reinhard Zumkeller, Oct 10 2013

			The lowest-order polynomial having points (1,2), (2,3), (3,5) and (4,7) is f(x) = 1/6 (-x^3 +9x^2 -14x +18). When evaluated at x = 5, f(5) = 8.

Jonathan Wellons, Table of n, a(n) for n = 1..1500

Cf. A082594, A140118.

a140119 = sum . a095195_row  -- Reinhard Zumkeller, Oct 10 2013

a(n) = sum(i=1, n, prime(i)*(-1)^(n-i)*binomial(n, i-1)); \\ Michel Marcus, Jun 28 2020

a(n) = Sum_{i=1..n} prime(i) * (-1)^(n-i) * C(n,i-1).

A082674 Constant term when a polynomial of degree n is fitted to the lower members of the first n+1 twin prime pairs.

Original entry on oeis.org

1, 5, 9, 19, 41, 87, 187, 425, 1041, 2689, 7031, 18015, 44503, 105503, 240267, 527035, 1116023, 2283321, 4509661, 8574251, 15613035, 26989459, 43596473, 63714861, 77517775, 54160583, -87072621, -539390369, -1742001769, -4661299497

Offset: 1

Cino Hilliard, May 19 2003

			A 5th-degree polynomial through the 6 points (1, 3), (2, 5), (3, 11), (4, 17), (5, 29), (6, 41) has constant term 41.

Cf. A001359, A082594, A082675.

A088460 := proc(n) local i,p ; i := 1 ; p := 0 ; while true do while ithprime(i+1)-ithprime(i) <> 2 do i := i+1 ; od ; p := p+1 ; if p = n then RETURN( ithprime(i) ) ; fi ; i := i+1 ; od ; end: A082674 := proc(n) local rhs,co, row,col; rhs := linalg[vector](n+1) ; co := linalg[matrix](n+1,n+1) ; for row from 1 to n+1 do rhs[row] := A088460(row) ; for col from 1 to n+1 do co[row,col] := row^(col-1) ; od ; od ; linalg[linsolve](co,rhs)[1] ; end: for n from 1 to 30 do printf("%d,",A082674(n)) ; od ; # R. J. Mathar, Oct 31 2006

a(n) = A082675(n) - 2.

Corrected and extended by R. J. Mathar, Oct 31 2006

A082675 Constant term when a polynomial of degree <= n is fitted to the first n+1 upper members of the twin prime pairs.

Original entry on oeis.org

3, 7, 11, 21, 43, 89, 189, 427, 1043, 2691, 7033, 18017, 44505, 105505, 240269, 527037, 1116025, 2283323, 4509663, 8574253, 15613037, 26989461, 43596475, 63714863, 77517777, 54160585, -87072619, -539390367, -1742001767, -4661299495

Offset: 1

Cino Hilliard, May 19 2003

			A 5th degree polynomial through the 6 points (1, 5), (2, 7), (3, 13), (4, 19), (5, 31), (6, 43) has constant term 43.

Robert Israel, Table of n, a(n) for n = 1..1000
Cino Hilliard, Sicurvqf.exe

Equals lower-member sequence (A082674) + 2.

Cf. A082594.

A006512 := proc(n) local i,p ; i := 1 ; p := 0 ; while true do while ithprime(i+1)-ithprime(i) <> 2 do i := i+1 ; od ; p := p+1 ; if p = n then RETURN( ithprime(i+1) ) ; fi ; i := i+1 ; od ; end: A082675 := proc(n) local rhs,co, row,col; rhs := linalg[vector](n+1) ; co := linalg[matrix](n+1,n+1) ; for row from 1 to n+1 do rhs[row] := A006512(row) ; for col from 1 to n+1 do co[row,col] := row^(col-1) ; od ; od ; linalg[linsolve](co,rhs)[1] ; end: for n from 1 to 30 do printf("%d,",A082675(n)) ; od ; # R. J. Mathar, Oct 31 2006

Corrected and extended by R. J. Mathar, Oct 31 2006

Definition edited by Robert Israel, Jun 14 2024

A293210 a(n) = [x^n] (1/(1 - x)^n)Sum_{k>=1} prime(k)x^k.

Original entry on oeis.org

0, 2, 7, 26, 97, 366, 1388, 5288, 20225, 77618, 298766, 1153018, 4460072, 17287558, 67129566, 261095420, 1016994627, 3966529870, 15488964428, 60549061804, 236932924494, 927984726826, 3637661249946, 14270586372348, 56024073085546, 220089137078792, 865154426179408, 3402841810234762, 13391422390407194

Offset: 0

Ilya Gutkovskiy, Oct 02 2017

nonn

Diagonal of A254858.
Cf. A000040, A007504, A014148, A014150, A178138, A254784.
CF. A007442, A082594.

Table[SeriesCoefficient[1/(1 - x)^n Sum[Prime[k] x^k, {k, 1, n}], {x, 0, n}], {n, 0, 28}]

a(n) = A254858(n,n).

cp's OEIS Frontend

A007442 Inverse binomial transform of primes.

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A140119 Extrapolation for (n + 1)-st prime made by fitting least-degree polynomial to first n primes.

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A082674 Constant term when a polynomial of degree n is fitted to the lower members of the first n+1 twin prime pairs.

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Maple

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A082675 Constant term when a polynomial of degree <= n is fitted to the first n+1 upper members of the twin prime pairs.

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Maple

Extensions

A293210 a(n) = [x^n] (1/(1 - x)^n)*Sum_{k>=1} prime(k)*x^k.

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A293210 a(n) = [x^n] (1/(1 - x)^n)Sum_{k>=1} prime(k)x^k.