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

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, 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, 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

Offset: 6320430

Author

Cino Hilliard, Dec 29 2003

Keywords

• easy
• cons
• nonn
• fini

Comments

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)).

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

Links

• Muniru A Asiru, Table of n, a(n) for n = 6320430..6321430
• GIMPS, List of Known Mersenne Prime Numbers
• OEIS Wiki, Mersenne Primes (with a list of similar sequences)

Crossrefs

Cf. A000043 (main entry), A000668, A028335 (lengths).

Programs

Mathematica

RealDigits[10^N[20996011Log[10, 2] - 6320430, 105]][[1]] (* Georg Fischer, Mar 19 2019 after Jakob Vecht in A117853 *)

PARI

\\ digits of the 40th Mersenne prime: 2^20996011 - 1
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",") ) }
digitsm40(105,p)

PARI

write("a089578.txt", 2^20996011 - 1) \\ Georg Fischer, Mar 18 2019

Extensions

Edited by Georg Fischer, Mar 19 2019