A135928 - OEIS

A135928

Digital roots of the Mersenne primes.

3, 7, 4, 1, 1, 4, 1, 1, 1, 4, 4, 1, 4, 1, 1, 1, 1, 1, 4, 1, 4, 4, 4, 4, 4, 1, 1, 4, 1, 1, 1, 4, 4, 1, 4, 4, 4, 4, 1, 1, 1, 4, 1, 4, 4, 1, 4, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

As a consequence of the fact that all prime numbers are of the form 6n-1 or 6n+1 for p>3, all the elements of this sequence after the second will be either 1 or 4, although there is no obvious pattern to their distribution. We can use this result to show that all Mersenne primes after the first are congruent to 1, modulo 6.

LINKS

Table of n, a(n) for n=1..48.

Syed Asadulla, Digital Roots of Mersenne Primes and Even Perfect Numbers, The College Mathematics Journal, Vol. 15, No. 1. (1984), pp. 53-54.

Eric Weisstein's World of Mathematics, Digital Root.

FORMULA

a(n) = A010888(A000668(n)).

For n > 2, a(n) = (A000043(n) mod 3)^2. - Jens Kruse Andersen, Jul 29 2014

EXAMPLE

The fourth Mersenne prime is 127, which has a digital root of 1. Hence a(4)=1.

MATHEMATICA

DigitalRoot[n_]:=FixedPoint[Plus@@IntegerDigits[ # ]&, n]; data1=Select[Range[4500], PrimeQ[2^#-1] &]; data2=2^#-1 &/@data1; DigitalRoot/@data2

CROSSREFS