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

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

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)

Examples

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

References

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

Links

• 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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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