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Digital roots of the Mersenne primes.
2

%I #24 Feb 23 2023 07:40:26

%S 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,

%T 4,4,4,4,1,1,1,4,1,4,4,1,4,4

%N Digital roots of the Mersenne primes.

%C 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.

%H Syed Asadulla, <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/digital-roots-of-mersenne-primes-and-even-perfect-numbers">Digital Roots of Mersenne Primes and Even Perfect Numbers</a>, The College Mathematics Journal, Vol. 15, No. 1. (1984), pp. 53-54.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>.

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

%F For n > 2, a(n) = (A000043(n) mod 3)^2. - _Jens Kruse Andersen_, Jul 29 2014

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

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

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

%K nonn,base,hard,more

%O 1,1

%A _Ant King_, Dec 07 2007

%E a(40)-a(43) (using A000043) from _Jens Kruse Andersen_, Jul 29 2014

%E a(44)-a(48) from mersenne.org added by _M Sayer_, Jan 05 2023