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A373307 Binary digits of Pi selected by stepping forward d+1 places at digit d, i.e., by skipping the next d places.

1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0

Offset: 1

Karl Levy, May 31 2024

Are the digits uniformly distributed? Are all digit sequences uniformly distributed?

			The sequence starts with the first digit of the binary expansion of Pi, which is 1. The next term is the digit 1+1 places after this, namely, 0, and so on.
The digits selected from Pi begin
  Pi=1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, ...
     ^     ^  ^  ^     ^  ^     ^  ^  ^  ^     ^

Cf. A004601.

Cf. A373079, A373304.

a={1}; s=1; For[n=2, n<=100, n++, s+=Part[a,n-1]+1; digits=First[RealDigits[Pi,2,s]]; AppendTo[a,Part[digits,s]]]; a

a(n) = the (n+a(1)+a(2)+...+a(n-1))-th digit in the binary expansion of Pi.

A373304 Decimal digits from Pi selected by stepping forward d+1 places at digit d, i.e., by skipping the next d places.

Original entry on oeis.org

3, 5, 5, 2, 4, 3, 2, 5, 1, 7, 7, 9, 0, 2, 9, 2, 4, 0, 6, 3, 4, 8, 5, 7, 8, 7, 0, 1, 5, 4, 4, 9, 2, 1, 9, 4, 4, 8, 3, 2, 2, 9, 6, 0, 2, 4, 3, 0, 2, 7, 6, 5, 8, 9, 9, 7, 6, 0, 6, 4, 5, 1, 6, 5, 6, 1, 5, 8, 3, 1, 9, 8, 9, 3, 8, 9, 3, 9, 7, 6, 8, 4, 2, 7, 2, 9, 7, 7, 7, 6, 6, 5, 0, 6, 7, 4, 7, 3, 1, 8

Offset: 1

Karl Levy, May 31 2024

Are the digits uniformly distributed? Are all consecutive digit subsequences uniformly distributed?

			The sequence starts with the first digit of the decimal expansion of Pi, which is 3. The next term is the digit 3+1 places after this, namely, 5, and so on.
The digits selected from Pi begin
  Pi = 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, ...
       ^           ^                 ^                 ^        ^

Cf. A000796.

Cf. A373079.

a={3}; s=1; For[n=2, n<=100, n++, s+=1+Part[a,n-1]; digits=First[RealDigits[Pi,10,s]]; AppendTo[a,Part[digits,s]]]; a

a(n) = the (n+a(1)+a(2)+...+a(n-1))-th digit in the decimal expansion of Pi.

A373079 Decimal digits of Pi selected by stepping forward d places at digit d or 10 places if d=0.

Original entry on oeis.org

3, 1, 5, 3, 9, 2, 4, 3, 9, 7, 3, 1, 0, 4, 3, 8, 6, 8, 2, 3, 1, 1, 7, 1, 4, 6, 2, 0, 4, 5, 2, 3, 2, 3, 4, 2, 4, 1, 7, 1, 0, 1, 1, 0, 2, 4, 4, 3, 9, 9, 4, 2, 4, 4, 3, 6, 5, 0, 8, 6, 1, 0, 2, 3, 3, 7, 1, 4, 3, 4, 0, 8, 5, 0, 0, 6, 9, 3, 3, 4, 0, 1, 4, 1, 9, 4, 5, 3, 7, 5, 1, 8, 9, 1, 0, 4, 3, 9, 3, 8

Offset: 1

Karl Levy, May 22 2024

Are the digits uniformly distributed?

			The sequence starts with the first digit of the decimal expansion of Pi, which is 3. The next term is the digit 3 places after this, namely, 1, and so on.
The digits selected from Pi begin
  Pi = 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, ...
       ^        ^  ^              ^        ^

Cf. A000796,

Cf. A373304, A373307.

a={3}; s=1; For[n=2, n<=100, n++, s+=Part[a,n-1]+10KroneckerDelta[Part[a,n-1]]; digits=First[RealDigits[Pi,10,s]]; AppendTo[a,Part[digits,s]]]; a (* Stefano Spezia, May 31 2024 *)

a(n) = the (1 + Sum_{i=1..n-1} a(i) + 10*delta(a(i),0))-th digit in the decimal expansion of Pi, where delta is the Kronecker symbol.

a(25)-a(100) from Stefano Spezia, May 31 2024

A137244 a(n) = lcm_{k=0..n} (k! + 1).

Original entry on oeis.org

2, 2, 6, 42, 1050, 127050, 13086150, 65967282150, 2659866783570150, 13594579130827036650, 4484729304047661947505150, 179016047168539016473835519025150, 85748973198421705721932588223712809265150, 533960639770963461900374948788827304744234574385150

Offset: 0

Karl Levy, Mar 09 2008

I came upon this sequence in an attempt to solve an open Erdős problem: Show that Sum_{k>=0} 1/(k!+1) is rational/irrational/transcendental.

Harvey P. Dale, Table of n, a(n) for n = 0..43

Cf. A038507, A000142.

With[{t=Range[0,20]!+1},Table[LCM@@Take[t,n],{n,Length[t]}]] (* Harvey P. Dale, Dec 21 2015 *)

a(n) = {lc = 1; for (k=0, n, lc = lcm(lc, k!+1);); return (lc);} \\ Michel Marcus, Jul 25 2013

a(n) = lcm_{k=0..n} (k! + 1).

More terms from Harvey P. Dale, Dec 21 2015

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User: Karl Levy

A373307 Binary digits of Pi selected by stepping forward d+1 places at digit d, i.e., by skipping the next d places.

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A373304 Decimal digits from Pi selected by stepping forward d+1 places at digit d, i.e., by skipping the next d places.

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A373079 Decimal digits of Pi selected by stepping forward d places at digit d or 10 places if d=0.

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A137244 a(n) = lcm_{k=0..n} (k! + 1).

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