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

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

Cf. A000668, A000043, A003010, A001566, A010888, A135927.

Sequence in context: A365729 A181912 A108297 * A011444 A272004 A010471

Adjacent sequences: A135925 A135926 A135927 * A135929 A135930 A135931

Keyword

nonn,base,hard,more

Author

Ant King, Dec 07 2007

Extensions

a(40)-a(43) (using A000043) from Jens Kruse Andersen, Jul 29 2014

a(44)-a(48) from mersenne.org added by M Sayer, Jan 05 2023

Status

approved