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A038194
Iterated sum-of-digits of n-th prime; or digital root of n-th prime; or n-th prime modulo 9.
48
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
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
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