A163057 - OEIS

A163057

An alternating sum from the n-th odd number up to the n-th odd prime.

2, 4, 6, 9, 11, 14, 16, 19, 23, 25, 29, 32, 34, 37, 41, 45, 47, 51, 54, 56, 60, 63, 67, 72, 75, 77, 80, 82, 85, 93, 96, 100, 102, 108, 110, 114, 118, 121, 125, 129, 131, 137, 139, 142, 144, 151, 158, 161, 163, 166, 170, 172, 178, 182, 186, 190, 192, 196, 199, 201, 207, 215

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OFFSET

1,1

COMMENTS

Define the alternating sum S(n) = -Sum_{j=0..n} (-1)^j*j = -A130472(n).

Then a(n) = S(n-th odd prime) - S((n-th odd number) - 1), as if the sum were ranging over all j from the n-th odd number up to the n-th odd prime.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = -A130472(A065091(n)) + A130472(A005408(n-1) - 1).

a(n) = (1/2)*(-1 + 2n + A000040(n + 1)). - David Trimas, Jul 21 2024

EXAMPLE

a(1) = 1 - 2 + 3 = 2;

a(2) = 3 - 4 + 5 = 4;

a(3) = 5 - 6 + 7 = 6;

a(4) = 7 - 8 + 9 - 10 + 11 = 9;

a(5) = 9 - 10 + 11 - 12 + 13 = 11;

a(6) = 11 - 12 + 13 - 14 + 15 - 16 + 17 = 14.

MAPLE

A130472 := proc(n) (-1)^n*floor((n+1)/2) ; end:

A005408 :=proc(n) 2*n+1 ; end:

A065091 :=proc(n) ithprime(n+1) ; end:

A163057 := proc(n) -A130472(A065091(n)) + A130472(A005408(n-1) -1) ; end: seq(A163057(n), n=1..80) ; # R. J. Mathar, Jul 27 2009

MATHEMATICA

1/2 (-1 + 2 # + Prime[# + 1]) & /@ Range[100] (* David Trimas, Jul 21 2024 *)

PROG

(Python)

from sympy import sieve as A000040

def A163057(n): return n + A000040[n + 1] // 2 # Karl-Heinz Hofmann, Jul 23 2024

CROSSREFS

Cf. A000040, A065091, A005408, A130472.

Sequence in context: A224995 A091626 A085223 * A242137 A138326 A274214

Adjacent sequences: A163054 A163055 A163056 * A163058 A163059 A163060

KEYWORD

nonn,easy

AUTHOR

Juri-Stepan Gerasimov, Jul 20 2009

EXTENSIONS

Rephrased in terms of A130472 by R. J. Mathar, Jul 27 2009

STATUS

approved