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A282782
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Numbers that are equal to a product of powers of digits where the exponents from left to right decrease with 1 and the exponent for the units digit is 1.
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0
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1715
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history;
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internal format)
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OFFSET
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1,3
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COMMENTS
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Up to 10^9 no other number matches the rule.
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LINKS
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EXAMPLE
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1 = 1^1, 2 = 2^1, ..., 1715 = (1^4)*(7^3)*(1^2)*(5^1).
These numbers do not match the rule:
46: (4^2)*(6^1) = 96 <> 46.
234: (2^3)*(3^2)*(4^1) = 288 <> 234.
4342: (4^4)*(3^3)*(4^2)*(2^1) = 221184 <> 4342.
46914: (4^5)*(6^4)*(9^3)*(1^2)*(4^1) = 3869835264 <> 46914.
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MATHEMATICA
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mx = 10^50; test[n_] := n == Times @@ (IntegerDigits[n] ^Reverse[Range@ IntegerLength@ n]); Union@Reap[Do[n = 2^i 3^j 7^k; If[test@n, Sow@n], {i, 0, Log2[mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[7, mx/2^i/3^j]}]; Do[n = 5 3^j 7^k; If[test@n, Sow@n], {j, 0, Log[3, mx/5]}, {k, 0, Log[7, mx/ 5/ 3^j]}]][[2, 1]] (* Search up to 10^50, Giovanni Resta, Feb 22 2017 *)
Select[Range[0, 2000], Times @@ MapIndexed[#1^First[#2] &, Reverse@ IntegerDigits@ #] == # &] (* Michael De Vlieger, Feb 22 2017 *)
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PROG
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(VBA) ' For example for 5-figure numbers:
Dim zahl As String
For i = 10000 To 99999
zahl = i
If i = CInt(Left(zahl, 1)) ^ 5 * CInt(Right(Left(zahl, 2), 1)) ^ 4 * CInt(Right(Left(zahl, 3), 1)) ^ 3 * CInt(Right(Left(zahl, 4), 1)) ^ 2 * CInt(Right(zahl, 1)) ^ 1 Then
MsgBox (i)
End If
Next i
(PARI) list(lim)=my(v=List([0]), t7, t37, t); for(a=0, logint(lim\1, 7), t7=7^a; for(b=0, logint(lim\t7, 3), t=t37=t7*3^b; while(t<=lim, if(is(t), listput(v, t)); t<<=1); t=t37; while(t<=lim, if(is(t), listput(v, t)); t*=5))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017
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CROSSREFS
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Subsequence of A002473; apart from the first term, a subsequence of A238985.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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