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
Examples
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)
Links
- Jon E. Schoenfield, Magma program
Programs
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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 *)
Extensions
a(6)-a(8) from Metin Sariyar, Feb 04 2021
More terms from Jon E. Schoenfield, Feb 05 2021