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A062853
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When expressed in base 3 and then interpreted in base 4, is a multiple of the original number.
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14
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0, 1, 2, 53, 91, 182, 194, 273, 546, 582, 948, 1092, 1236, 2184, 2527, 9373, 19238, 28119, 57714, 84357, 173142, 185640, 452807, 21774372, 48833136, 65323116, 1145127998, 3435383994, 4804366457, 11296002941, 14224061544, 18500792316, 28413081060, 33888008823
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OFFSET
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1,3
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COMMENTS
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Let u(k) be the result of expressing an integer k in base 3 and interpreting the result as a base-4 number, and define the ratio r(k) = u(k)/k. Then (after the initial term 0) the sequence consists of the integers k > 0 such that r(k) is an integer.
Note that, among all numbers k in any interval [m*3^j, (m+1)*3^j - 1] where m > 0, r(k) is maximized at k = m*3^j and minimized at (m+1)*3^j - 1. Consequently, there cannot be any terms in that interval unless there is at least one integer in the interval [r((m+1)*3^j - 1), r(m*3^j)]. (This observation is implemented in the Magma program below, which, when run on the Magma Calculator, computes the first 34 terms in about 0.5 seconds.) (End)
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LINKS
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EXAMPLE
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53 = 1222_3; 1222_4 = 106 = 2*53.
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MATHEMATICA
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fQ[n_] := Mod[ FromDigits[ IntegerDigits[n, 3], 4], n] == 0;
k = 1; lst = {};
While[k < 10^10/8, If[ fQ@k, AppendTo[ lst, k]; Print@k]; k++ ];
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PROG
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(Magma)
N := 34; // max # of terms
A := [0];
D := [1]; // base-3 dgts (reversed) at curr srch point
j := 1; // pointer (at ones place)
while #A lt N do
if j eq 1 then // test a single integer (k)
k := Seqint(D, 3);
if Seqint(D, 4) mod k eq 0 then
A[#A+1] := k;
end if;
D[j] +:= 1;
else // test the interval [k0, k1]
k0 := Seqint(D, 3);
k1 := k0 + 3^(j - 1) - 1;
u0 := Seqint(D, 4);
u1 := Seqint(Intseq(k1, 3), 4);
if u0 div k0 gt (u1 - 1) div k1 then
// at least 1 integer in interval [u1/k1, u0/k0]
j -:= 1; // test its 3 subintervals
else
D[j] +:= 1;
end if;
end if;
while D[j] eq 3 do // all 3 subintervals tested
D[j] := 0; // reset
j +:= 1; // move up to larger interval
if j gt #D then
D[j] := 1; // add a digit
break;
end if;
D[j] +:= 1;
end while;
end while;
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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Offset changed to 1 and a(28), a(29) from Georg Fischer, Mar 03 2023
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STATUS
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approved
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