A394263 - OEIS

A394263

Positive integers k for which the Euclidean distance between the vector of proportions of the first k decimal digits of Pi and the uniform distribution (1/10, ..., 1/10) sets a new minimum.

1, 2, 3, 5, 6, 7, 8, 12, 13, 14, 20, 21, 22, 23, 24, 37, 38, 39, 40, 41, 42, 55, 58, 69, 70, 71, 72, 73, 74, 75, 76, 78, 99, 100, 122, 132, 133, 134, 157, 159, 160, 182, 183, 185, 243, 244, 246, 247, 249, 254, 308, 309, 310, 312, 316, 317, 321, 352, 359, 361

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OFFSET

1,2

COMMENTS

Conjecture: The digit proportions of an initial segment of the decimal expansion of Pi can get arbitrarily close to, but not equal to, the uniform distribution. This implies that there are infinitely many terms in this sequence and in the related sequences A394264 and A394265.

The corresponding sequences for bases 2, 3, 4, and 5 are finite, because each digit occurs equally many times among the first 4, 15, 4, and 75 digits of Pi in base 2 (see A039624), 3 (see A278977, A278978, and A278979), 4, and 5 respectively:

base 2: 1, 3, 4;

base 3: 1, 2, 6, 7, 11, 12, 13, 14, 15;

base 4: 1, 2, 3, 4;

base 5: 1, 2, 4, 8, 9, 11, 18, 19, 20, 21, 22, 26, 27, 28, 30, 41, 43, 46, 47, 53, 54, 56, 66, 67, 68, 74, 75.

LINKS

Pontus von Brömssen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to the number Pi.

EXAMPLE

| | frequencies f_i of 0-9 in first a(n) digits |

n | a(n) | 0 1 2 3 4 5 6 7 8 9 | Sum (f_i/a(n)-1/10)^2

---+------+---------------------------------------------+-----------------------

1 | 1 | 0 0 0 1 0 0 0 0 0 0 | 9/10 = 0.9