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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.
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:
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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...
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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)
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