A121280 -id:A121280 - cp's OEIS frontend

A053746 Positions of '2's in the decimal expansion of Pi, where positions 1, 2, 3, ... correspond to digits 3, 1, 4, ...

7, 17, 22, 29, 34, 54, 64, 74, 77, 84, 90, 94, 103, 113, 115, 136, 137, 141, 150, 161, 166, 174, 186, 187, 204, 222, 230, 242, 245, 261, 276, 281, 290, 293, 299, 303, 327, 330, 334, 336, 338, 355, 375, 381, 407

Offset: 1

Simon Plouffe, Feb 20 2000

See A037001 for the variant where digits 3, 1, 4, ... correspond to positions 0, 1, 2, ... - M. F. Hasler, Jul 28 2024

			Pi = 3.1415926... where the first '2' occurs as the 7th digit.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the number Pi

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A037001 (= a(n) - 1: the same with different offset).
Cf. A053745 - A053753 (similar for digits 1 through 9).
Cf. A035117 (first occurrence of at least n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
Cf. A096755 (first occurrence of exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).
Cf. A121280 = A068987 - 1: position of "123...n" in Pi's decimals.
Cf. A176341: first occurrence of n in Pi's digits.
Cf. A088566 (primes in this sequence).

Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 2]] (* Vincenzo Librandi, Oct 07 2013 *)

A053746_upto(N=999)={localprec(N+20); select(d->d==2, digits(Pi\10^-N), 1)} \\ M. F. Hasler, Jul 28 2024

a(n) = A037001(n) + 1. - Georg Fischer, May 31 2021

Changed offset from 0 to 1 by Vincenzo Librandi, Oct 07 2013

A307581 Position of the first permutation of { 0 .. n-1 } occurring in the digits of Pi written in base n.

Original entry on oeis.org

0, 2, 0, 6, 15, 5, 371, 742, 60, 787

Offset: 2

M. F. Hasler, Apr 15 2019

"The first permutation of {0 .. n-1}" means the first string of n distinct digits.
"Position" means the index of the digit where this string begins, where index = p means the digit corresponding to n^-p: e.g., the first digit after the decimal point would have index 1.
By inspection, a(12) > 1000. - Alvin Hoover Belt, Mar 17 2021

			Pi written in base 2 is 11.0...[2] so "10" occurring at position a(2) = 0 (digits corresponding to 2^0 and 2^-1) is the first permutation of the digits 01 to occur in the digits of Pi written in base 2.
Pi written in base 3 is 10.0102...[3], so "102" occurring at position a(3) = 2 (the string starts at the digit corresponding to 3^-2) is the first permutation of digits 012 to occur in the digits of Pi written in base 3.
Pi written in base 4 is 3.021...[4], so "3021" occurring at position a(4) = 0 (the string starts at the digit corresponding to 4^0) is the first permutation of digits 0123 to occur in the digits of Pi written in base 4.
Pi written in base 5 is 3.0323221430...[5], so "21430" occurring at position a(5) = 6 (the string starts at the digit corresponding to 5^-6) is the first permutation of digits 01234 to occur in the digits of Pi written in base 5.
Pi = 3.141592653589793238462643383279502884197169399375105820974944592307816... (in base 10) has the first string of 10 distinct digits, "4592307816", starting at position a(10) = 60.

Cf. A307582 (start of first occurrence of (0, ..., n-1) in digits of Pi in base n).
Cf. A307583 (start of last permutation of {0 .. n-1} not to occur earlier, in base-n digits of Pi).
Cf. A068987, A121280.

A307581(n,x=Pi,m=n^n)=for(k=0,oo,#Set(d=digits(x\n^-k%m,n))>=n && (#Set(d)==n||vecsort(d)==[1..n-1]) && return([k-n+1,digits(x\n^-k,n)])) \\ Returns position and the digits up to there. Ensure sufficient realprecision (\p): an error should occur if a suitable permutation of digits is not found early enough, but in case of results near the limit of precision, it is suggested to double check (by increasing the precision further) that the relevant digits are all correct.

a(n) <= A307582(n) <= A307583(n).

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

Original entry on oeis.org

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

Offset: 2

M. F. Hasler, Apr 15 2019

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.

			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.

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.

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.

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

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

A307583 Position where the last of all n! permutations of { 0 .. n-1 } occurs in the digits of Pi written in base n.

Original entry on oeis.org

2, 82, 961, 15136

Offset: 2

M. F. Hasler, Apr 15 2019

By "permutation of { 0 .. n-1 }" we mean a string of n distinct digits. "The last" means the permutation which occurs for the first time later than all other permutations.

Position = k means that the string starts with the digit corresponding to the weight n^-k; e.g., the first digit after the decimal point has position 1.

			Pi written in base 2 is 11.001...[2], so the first "10" occurs at position 0 (starting with the digit of units) and "01" occurs later at position a(2) = 2.
Pi written in base 3 is 10.010211012...[3], we see that the first permutation of 0..2 to appear is "102", at position 2; then "021" at position 3, then "012" at position 7, then "201" at position 12, then "120" at position 39, and finally "210", the last partition not occurring earlier, at position 82 = a(3).
Pi written in base 4 is 3.02100333...[4]; the first permutation of 0..3 is "3012" at position 0 (starting at units digit '3'), the next distinct permutation to occur is "2031" at position 27 etc.; the last permutation not to occur earlier is "2310" at position 961 = a(4).

Cf. A307581 (first start of any permutation of 0 .. n-1 in base-n digits of Pi).
Cf. A307582 (first occurrence of "01...(n-1)" in digits of Pi written in base n).
Cf. A068987 (occurrence of 123...n in decimal digits of Pi), A121280.

A307583(n,x=Pi,m=n^n,S=[])={for(k=n-2,oo, #Set(d=digits(x\n^-k%m,n)) < n-1 && next; #Set(d)==n || vecsort(d)==[1..n-1] || next; setsearch(S,d) && next; printf("%d: %d, ",k-n+1,Vec(d,-n));S=setunion(S,[d]);#S==n!&&return(k-n+1))}

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A053746 Positions of '2's in the decimal expansion of Pi, where positions 1, 2, 3, ... correspond to digits 3, 1, 4, ...

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A307581 Position of the first permutation of { 0 .. n-1 } occurring in the digits of Pi written in base n.

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

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A307583 Position where the last of all n! permutations of { 0 .. n-1 } occurs in the digits of Pi written in base n.

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Author

Keywords

Comments

Examples

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Programs

PARI