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

1,1

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

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004

Formula

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

Example

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.

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 *)

Prog

(PARI) forprime(p=2, 600, print1(p%9, ", "))
(Magma) [p mod 9: p in PrimesUpTo(500)]; // Vincenzo Librandi, May 06 2014
(Haskell)
a038194 = flip mod 9 . a000040  -- Reinhard Zumkeller, Dec 10 2014

Crossrefs

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

Sequence in context: A337007 A074464 A074463 * A111309 A007605 A077765

Adjacent sequences: A038191 A038192 A038193 * A038195 A038196 A038197

Keyword

nonn,base,easy

Author

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

Extensions

Edited by Klaus Brockhaus, Feb 16 2002

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

Status

approved