Pickover's Sequence

Pickover's sequence gives the starting positions in the decimal expansion of  (ignoring the leading 3) in which the first  digits of  occur (counting the leading 2). So, since , the first digit "2" of  occurs at position 6. Continuing, the sequence is given by 6, 28, 241, 11706, 28024, 33789, 1526800, 73154827, ... (OEIS A090898).

Conversely, consider the sequence formed by the expansion of  (ignoring the leading 2) in which the first  digits of  occur (counting the leading 3). So, since , the first digit "3" of  occurs at position 17. Continuing, the sequence is given by 17, 189, 856, 17947, 53238, 1436935, 5000482, ... (OEIS A115234).

REFERENCES:

Anderson, D. "The Pi-Search Page." http://www.angio.net/pi/piquery.

Pickover, C. A. "A Connection between  and ." Ch. 99 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 206-208 and 337-339, 2002.

Sloane, N. J. A. Sequences A090898 and A115234 in "The On-Line Encyclopedia of Integer Sequences."