cp's OEIS Frontend

A089578 Decimal expansion of 2^20996011 - 1, the 40th Mersenne prime A000668(40).

%I A089578 #24 Jun 04 2021 10:51:56
%S A089578 1,2,5,9,7,6,8,9,5,4,5,0,3,3,0,1,0,5,0,2,0,4,9,4,3,0,9,5,7,4,8,2,4,3,
%T A089578 1,1,4,5,5,9,9,3,4,1,6,0,8,5,3,5,1,8,3,5,9,5,2,2,5,4,6,7,0,1,2,5,6,5,
%U A089578 4,9,8,7,6,8,9,0,8,3,5,1,5,6,0,2,2,1,2,4,0,0,9,6,8,0,2,8,2,8,5,3,6,1,3,2,5
%N A089578 Decimal expansion of 2^20996011 - 1, the 40th Mersenne prime A000668(40).
%C A089578 We can compute the digits of 2^p directly by noting that 2^p = 10^(p*log(2)/log(10)) = 10^(p*log_10(2)). This result is 10^(i+f) where i is the integer part and f the fractional part. Then 10^f will produce a decimal number i.d1d2d3d4... where i is an integer from 1 to 9 (zero cannot occur in i) and d1, d2 ... are the digits in the fractional part where 0 is allowed. So i is the first digit in 2^p, d1 the second, d2 the third etc. The expansion is self evident in the PARI program. This routine allows the direct computation of the digits of any base to a power: k^p = 10^(p*log_10(k)).
%C A089578 The 40th Mersenne prime found by GIMPS / Michael Shafer in 2003 is 1259768954503301...4065762855682047 = 2^20996011 - 1. The second PARI program below computes all digits. - _Georg Fischer_, Mar 18 2019
%H A089578 Muniru A Asiru, <a href="/A089578/b089578.txt">Table of n, a(n) for n = 6320430..6321430</a>
%H A089578 GIMPS, <a href="https://www.mersenne.org/primes/">List of Known Mersenne Prime Numbers</a>
%H A089578 OEIS Wiki, <a href="https://oeis.org/wiki/Mersenne_primes">Mersenne Primes</a> (with a list of similar sequences)
%t A089578 RealDigits[10^N[20996011Log[10, 2] - 6320430, 105]][[1]] (* _Georg Fischer_, Mar 19 2019 after Jakob Vecht in A117853 *)
%o A089578 (PARI) \\ digits of the 40th Mersenne prime: 2^20996011 - 1
%o A089578 p = 20996011; digitsm40(n, p) = { default(realprecision,n); p10 = frac(p*log(2)/log(10)); v = 10^p10; for(j=1,n, d=floor(v); v=frac(v)*10; print1(d",") ) }
%o A089578 digitsm40(105,p)
%o A089578 (PARI) write("a089578.txt", 2^20996011 - 1) \\ _Georg Fischer_, Mar 18 2019
%Y A089578 Cf. A000043 (main entry), A000668, A028335 (lengths).
%K A089578 easy,cons,nonn,fini
%O A089578 6320430,2
%A A089578 _Cino Hilliard_, Dec 29 2003
%E A089578 Edited by _Georg Fischer_, Mar 19 2019