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

Author

Ant King, Dec 07 2007

Keywords

• nonn
• base
• hard
• more

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.

Examples

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Formula

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

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

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