A307582 Position of the first occurrence of (0, 1, ..., n-1) in the digits of Pi written in base n.

2, 7, 188, 2264, 27931, 110808, 23489363, 97438020

Offset: 2

Author

M. F. Hasler, Apr 15 2019

Keywords

• nonn
• base
• more
• hard

Comments

Position refers to the digit where there required sequence (0, ..., n-1) starts. Position = k means the digit '0' occurs as digit corresponding to the weight n-^k (and thereafter, the digit '1' will correspond to n^-(k+1) etc): e.g., the first digit after the decimal point has position 1.

Examples

			Pi written in base 2 is 11.001...[2], so the first "01" occurs at position a(2) = 2.
Pi written in base 3 is 10.010211012...[3], we see that the first occurrence of the string "012" is at position a(3) = 7.
Pi written in base 4 is 3.02100333...[4]; the string of digits "0123" does not occur until position a(4) = 188.
		

Crossrefs

Cf. A307581 (first occurrence of any permutation of 0 .. n-1, in base-n digits of Pi).

Cf. A307583 (start of last permutation of {0 .. n-1} not to occur earlier, in base-n digits of Pi).

Cf. A068987 (occurrence of 123...n in decimal digits of Pi), A121280.

Programs

PARI

A307582(n,x=Pi,m=Mod(sum(i=1,n-1,i*n^(n-1-i)),n^n))={for(k=0,oo,x\n^-k==m&&return(k-n+1))} \\ Ensure sufficient precision of the argument x = Pi.

Formula

A307581(n) <= a(n) <= A307583(n).

Extensions

a(7)-a(9) from Chai Wah Wu, Apr 07 2020