A199339 a(n) = number of primes with an even digit sum among the first n primes minus the number with an odd digit sum.

1, 0, -1, -2, -1, 0, 1, 2, 1, 0, 1, 2, 1, 0, -1, 0, 1, 0, -1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 5, 4, 5, 4, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 0, -1, -2, -1, -2, -3, -4, -3, -2, -3, -2, -1, -2, -3, -2, -1, -2, -3, -2, -1, -2, -3, -4, -5, -6, -5, -4, -5, -6, -5, -6

Offset: 1

Author

M. F. Hasler, Nov 14 2011

Keywords

• sign
• base

Examples

			a(1)=1 because the first prime has an even sum of digits.
a(2)=0, a(3)=-1, a(4)=-2 because the following primes (3,5,7) have odd sum of digits.
a(5)=-1, a(6)=0, a(7)=1, a(8)=2 because the 5th, 6th, 7th and 8th prime (11, 13, 17, 19) have an even sum of digits.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• CNRS Press release, The sum of digits of prime numbers is evenly distributed, May 12, 2010.
• Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
• ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 12, 2010.

Crossrefs

Cf. A007605, A007953, A130911, A200259, A200260, A119449, A119450, A200261, A200262, A200263, A200264.

Programs

Mathematica

a[1] := 1; a[n_] := a[n] = a[n - 1] + (-1)^(Plus@@IntegerDigits[Prime[n]]); Table[a[n], {n, 74}] (* Alonso del Arte, Nov 14 2011 *)

PARI

s=0;vector(90,n,s+=(-1)^A007953(prime(n)))

Formula

a(n)=sum_{k=1...n} (-1)^A007605(n).

Equals A200262 - A200264.