A341046 a(n) is the smallest k such that the fractional part of the decimal expansion of k*Pi begins with n zeros.

1, 36, 106, 29486, 32876, 66317, 1360120, 22060516, 78256779, 1151791169, 6701487259, 6701487259, 1142027682075, 2851718461558, 91822653867264, 136308121570117, 1952799169684491, 21208174623389167, 842468587426513207, 842468587426513207, 84383735478118508040

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Jon E. Schoenfield, Magma program

Example

a(0)=1 because Pi*1=3.1415... has 0 zeros at the start of the fractional part of the decimal expansion, and 1 is the smallest positive integer that has this property.
a(1)=36 because Pi*36=113.09733... has 1 zero, and 36 is the smallest positive integer that has this property.
a(2)=106 because Pi*106=333.00882128... has 2 zeros, and 106 is the smallest positive integer that has this property.
From Jon E. Schoenfield, Feb 05 2021: (Start)
For each term a(n), the integer part of the corresponding product Pi*a(n) is A341047(n). Terms and their corresponding untruncated products begin as follows:
.
   n       a(n)                 Pi*a(n)
  --  -------------  ------------------------------
   0              1              3.1415926535897...
   1             36            113.0973355292325...
   2            106            333.0088212805180...
   3          29486          92633.0009837486434...
   4          32876         103283.0000794180425...
   5          66317         208341.0000081143181...
   6        1360120        4272943.0000005495794...
   7       22060516       69305155.0000000911737...
   8       78256779      245850922.0000000061180...
   9     1151791169     3618458675.0000000005971...
  10     6701487259    21053343141.0000000000017...
  11     6701487259    21053343141.0000000000017...
  12  1142027682075  3587785776203.0000000000003...
.
a(10) = 6701487259 because no multiple of Pi less than the product 6701487259*Pi = 21053343141.0000000000017... has a fractional part whose first 10 digits after the decimal point are all zeros, but that product does.
Note, however, that that product's 11th digit after the decimal point is also a zero; thus, a(11) = a(10). Presumably, a similar situation occurs infinitely many times; a(n) = a(n-1) at n = 11, 19, 39, 41, 74, 156, 183, 217, 218, 219, 220, 247, .... The consecutive integers 217..220 are in this list because a(220)=a(219)=a(218)=a(217)=a(216).
(End)

Mathematica

A341046[n_] := Module[{m = 1, i = 0}, While[i < n + 1, i = Abs[Floor[Log[10, Abs[FractionalPart[N[Pi*m]]]]]]; m++]; m - 1]; Table[A341046[n], {n, 0, 7}] (* Robert P. P. McKone, Feb 04 2021 *)

Crossrefs

Cf. A000796, A341047.

Sequence in context: A182677 A076768 A282853 * A162940 A250426 A176685

Adjacent sequences: A341043 A341044 A341045 * A341047 A341048 A341049

Keyword

nonn,base

Author

Talha Ali, Feb 04 2021

Extensions

a(6)-a(8) from Metin Sariyar, Feb 04 2021

More terms from Jon E. Schoenfield, Feb 05 2021

Status

approved