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%I #47 Feb 18 2021 01:36:20
%S 0,1,2,3,4,5,6,7,8,9,1715
%N 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.
%C Up to 10^9 no other number matches the rule.
%C There are no other terms up to 10^200; this list is probably complete. - _Charles R Greathouse IV_, Feb 22 2017
%e 1 = 1^1, 2 = 2^1, ..., 1715 = (1^4)*(7^3)*(1^2)*(5^1).
%e These numbers do not match the rule:
%e 46: (4^2)*(6^1) = 96 <> 46.
%e 234: (2^3)*(3^2)*(4^1) = 288 <> 234.
%e 4342: (4^4)*(3^3)*(4^2)*(2^1) = 221184 <> 4342.
%e 46914: (4^5)*(6^4)*(9^3)*(1^2)*(4^1) = 3869835264 <> 46914.
%t 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 *)
%t Select[Range[0, 2000], Times @@ MapIndexed[#1^First[#2] &, Reverse@ IntegerDigits@ #] == # &] (* _Michael De Vlieger_, Feb 22 2017 *)
%o (VBA) ' For example for 5-figure numbers:
%o Dim zahl As String
%o For i = 10000 To 99999
%o zahl = i
%o 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
%o MsgBox (i)
%o End If
%o Next i
%o (PARI) is(n)=my(d=digits(n)); prod(i=1,#d,d[#d+1-i]^i)==n || !n \\ _Charles R Greathouse IV_, Feb 22 2017
%o (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
%Y Subsequence of A002473; apart from the first term, a subsequence of A238985.
%K nonn,base
%O 1,3
%A _Shmelev Aleksei_, Feb 21 2017
%E Leading 0 prepended by _David A. Corneth_, Feb 22 2017
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