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

6320430,2

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)

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A018878 A021389 A198429 * A020852 A053477 A104956

Adjacent sequences: A089575 A089576 A089577 * A089579 A089580 A089581

Keyword

easy,cons,nonn,fini

Author

Cino Hilliard, Dec 29 2003

Extensions

Edited by Georg Fischer, Mar 19 2019

Status

approved