Search: a121280 -id:a121280
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A053746
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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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+10
7
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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
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OFFSET
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1,1
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COMMENTS
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See A037001 for the variant where digits 3, 1, 4, ... correspond to positions 0, 1, 2, ... - M. F. Hasler, Jul 28 2024
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LINKS
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FORMULA
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EXAMPLE
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Pi = 3.1415926... where the first '2' occurs as the 7th digit.
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MATHEMATICA
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Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 2]] (* Vincenzo Librandi, Oct 07 2013 *)
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PROG
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(PARI) A053746_upto(N=999)={localprec(N+20); select(d->d==2, digits(Pi\10^-N), 1)} \\ M. F. Hasler, Jul 28 2024
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CROSSREFS
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Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A037001 (= a(n) - 1: the same with different offset).
Cf. A176341: first occurrence of n in Pi's digits.
Cf. A088566 (primes in this sequence).
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A307581
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Position of the first permutation of { 0 .. n-1 } occurring in the digits of Pi written in base n.
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+10
3
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OFFSET
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2,2
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COMMENTS
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"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.
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LINKS
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FORMULA
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EXAMPLE
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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
3: [2,
Pi written in base 3 = 1.00102...[3], so "102" occurring at position a(3) = 3 (the string starts at the digit corresponding to 3^-3) 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 written in base 6 is 3.0, 5, 0, 3, 3, 0, 0, 5, 1, 4, 1, 5, 1, 2, 4, 1, 0, 5, 2, 3...[6], so "102" occurring at position a(6) = 15 (the string starts at the digit corresponding to 3^-3) is the first permutation of digits 012 to occur in the digits of Pi written in base 3.
Pi written in base 7 is 3.06636514320...[7], so "6514320" occurring at position a(7) = 5 (the string starts at the digit corresponding to 7^-5) is the first permutation of digits 0123456 to occur in the digits of Pi written in base 3.
Pi written in base 8 is 3.11037...(360 digits omitted)...6253510756243...[8], so "10756243" occurring at position a(8) = 371 (the string starts at the digit corresponding to 8^-371) is the first permutation of digits 01234567 to occur in the digits of Pi written in base 3.
Pi written in base 9 has the first string of 9 distinct digits, "352710468", starting at position a(9) = 742.
Pi = 3.141592653589793238462643383279502884197169399375105820974944592307816... in base 10) has the first string of 10 distinct digits, "4592307816", starting at position a(10) = 60.
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PROG
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(PARI) 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.
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CROSSREFS
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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).
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KEYWORD
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nonn,more,base
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AUTHOR
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STATUS
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approved
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A307582
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Position of the first occurrence of (0, 1, ..., n-1) in the digits of Pi written in base n.
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+10
3
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OFFSET
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2,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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.
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PROG
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(PARI) A307582(n, x=Pi, m=Mod(sum(i=1, n-1, i*n^(n-1-i)), n^n))={for(k=oo, x\n^-k==m&&return(k-n+1)) \\ Ensure sufficient precision of the argument x = pi.
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CROSSREFS
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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).
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KEYWORD
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nonn,base,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A307583
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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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+10
3
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OFFSET
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2,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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PROG
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(PARI) 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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CROSSREFS
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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).
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KEYWORD
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nonn,base,more
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AUTHOR
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STATUS
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approved
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