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Numbers n such that n and n-th prime have only one common digit = 0.
+10
4
80, 140, 305, 306, 308, 409, 440, 504, 540, 600, 670, 704, 790, 807, 820, 902, 904, 906, 1011, 1012, 1013, 1014, 1016, 1017, 1072, 1096, 2002, 2025, 2033, 2043, 2052, 2065, 2066, 2067, 2070, 2073, 2097
COMMENTS
Other cases of common digit d: A107932 (d=1), A107933 (d=2), A107934 (d=3), A107935 (d=4), A107936 (d=5), A107937 (d=6), A107938 (d=7), A107939 (d=8), A107940 (d=9). Cf. A107930 - smallest m's for digits 0,...,9.
EXAMPLE
a(1)=80 because 80th prime, 409 and 80 have the only common digit = 0 and 80 is the smallest such a number.
MATHEMATICA
bb={}; Do[If[IntegerDigits [n]\[Intersection]IntegerDigits [Prime[n]]\[Equal]{0}, bb=Append[bb, n]], {n, 3000}]; bb
Select[Range[2200], Intersection[IntegerDigits[#], IntegerDigits[ Prime[#]]] == {0}&] (* Harvey P. Dale, Dec 18 2013 *)
Numbers n such that n and n-th prime have only one common digit = 3.
+10
4
23, 30, 32, 33, 34, 38, 63, 83, 103, 130, 131, 132, 143, 153, 235, 238, 311, 314, 330, 333, 338, 341, 343, 344, 345, 346, 349, 353, 354, 355, 356, 357, 360, 361, 366, 368, 370, 371, 378, 396, 399, 432, 433, 434, 435, 438, 439, 443, 453, 463, 473, 513, 523
COMMENTS
Other cases of common digit d: A107931 (d=0), A107932 (d=1), A107933 (d=2), A107935 (d=4), A107936 (d=5), A107937 (d=6), A107938 (d=7), A107939 (d=8), A107940 (d=9). Cf. A107930 - smallest m's for digits 0,...,9.
EXAMPLE
a(1)=23 because 23rd prime, 83 and 23 have 3 as their only common digit, and 23 is the smallest such number.
MAPLE
filter:= n -> convert(convert(n, base, 10), set) intersect convert(convert(ithprime(n), base, 10), set) = {3}:
MATHEMATICA
bb={}; Do[If[IntegerDigits [n]\[Intersection]IntegerDigits [ Prime[n]]\[Equal]{3}, bb=Append[bb, n]], {n, 1800}]; bb
Numbers n such that n and n-th prime have only one common digit = 1.
+10
3
11, 13, 18, 31, 41, 81, 100, 112, 113, 114, 115, 121, 125, 126, 128, 133, 135, 141, 152, 156, 157, 160, 164, 167, 171, 174, 175, 176, 177, 178, 179, 182, 184, 185, 186, 188, 190, 191, 192, 193, 194, 195, 197, 198, 199
COMMENTS
Other cases of common digit d: A107931 (d=0), A107933 (d=2), A107934 (d=3), A107935 (d=4), A107936 (d=5), A107937 (d=6), A107938 (d=7), A107939 (d=8), A107940 (d=9). Cf. A107930 - smallest m's for digits 0,...,9.
EXAMPLE
a(1)=11 because 11th prime, 31 and 11 have the only common digit = 1 and 11 is the smallest such a number.
MATHEMATICA
bb={}; Do[If[IntegerDigits [n]\[Intersection]IntegerDigits [ Prime[n]]\[Equal]{1}, bb=Append[bb, n]], {n, 200}]; bb
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