A038194 Iterated sum-of-digits of n-th prime; or digital root of n-th prime; or n-th prime modulo 9.

2, 3, 5, 7, 2, 4, 8, 1, 5, 2, 4, 1, 5, 7, 2, 8, 5, 7, 4, 8, 1, 7, 2, 8, 7, 2, 4, 8, 1, 5, 1, 5, 2, 4, 5, 7, 4, 1, 5, 2, 8, 1, 2, 4, 8, 1, 4, 7, 2, 4, 8, 5, 7, 8, 5, 2, 8, 1, 7, 2, 4, 5, 1, 5, 7, 2, 7, 4, 5, 7, 2, 8, 7, 4, 1, 5, 2, 1, 5, 4, 5, 7, 8, 1, 7, 2, 8

Offset: 1

Author

Den Roussel (DenRoussel(AT)webtv.net) and Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Integers with iterated sum-of-digits 3, 6 or 9 are divisible by 3, so 3 is the only prime with iterated sum-of-digits 3 and there are no primes with iterated sum-of-digits 6 or 9.

The remaining values are very evenly distributed: these are the number of appearances in the first 1007933 primes: 1:167878; 2:168079; 4:167984; 5:168027; 7:167906; 8:168058. - Carmine Suriano, Jun 22 2015

Asymptotically, the ratios (number of primes <= n and == i mod 9)/(number of primes <= n and == j mod 9) go to 1 as n -> infinity for all i,j in {1,2,4,5,7,8} by the Prime Number Theorem for Arithmetic Progressions. For more detailed analysis, see the Granville-Martin link. - Robert Israel, Jul 08 2015

Examples

			Prime(5) = 11, 1 + 1 = 2 hence a(5) = 2.
a(297)=7 because the 297th prime is 1951 and 1+9+5+1 = 16 -> 1+6 = 7.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004

Crossrefs

Cf. A007605, A010888, A061237 - A061242, A139413, A153110.

Programs

Haskell

a038194 = flip mod 9 . a000040  -- Reinhard Zumkeller, Dec 10 2014

Magma

[p mod 9: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014

Maple

A038194 := proc(n) return ithprime(n) mod 9: end: seq(A038194(n), n=1..100); # Nathaniel Johnston, May 04 2011

Mathematica

Table[Mod[Prime[n], 9], {n, 200}]
Mod[Prime[Range[100]], 9] (* Vincenzo Librandi, May 06 2014 *)

PARI

forprime(p=2,600,print1(p%9,","))

Formula

a(n) = A010888(A000040(n)).

Sum_k={1..n} a(k) ~ (9/2)*n. - Amiram Eldar, Dec 11 2024

Extensions

Edited by Klaus Brockhaus, Feb 16 2002

Edited at the suggestion of R. J. Mathar by N. J. A. Sloane, May 14 2008