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A115518
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Numbers n such that the sum of the digits of n times the sum of the digits squared of n equals n.
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1
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OFFSET
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1,2
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COMMENTS
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(a_1 + a_2 + .. + a_k)*(a_1^2 + a_2^2 + .. + a_k^2) = n, the k-digit number a_k...a_2a_1, in base 10.
The sequence is finite and all terms are listed. Proof. Each natural number n has not more than log_10(n)+1 digits. Furthermore each digit is, of course, not bigger than 9. This gives the inequality (a1+a2+..ap)(a1^2+a2^2+..+ap^2) <= 9*(log_10(n)+1)*81*(log_10(n)+1) = 729*(log_10(n)+1)^2. On the other hand, for all n > 20582 we have 729*(log_10(n)+1)^2 < n. Therefore all terms of the sequence have to be smaller than this upper bound. A simple computer search shows that indeed all terms are listed. - Stefan Steinerberger
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LINKS
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Table of n, a(n) for n=1..7.
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EXAMPLE
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(2+1+9+6)(2^2+1^2+9^2+6^2)=2196
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MAPLE
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p := proc (K) options operator, arrow; floor((log[10])(K))+1 end proc; A := proc (K) options operator, arrow; convert(K, base, 10) end proc; g := proc (K) options operator, arrow; sum(A(K)[i], i = 1 .. p(K)) end proc p := proc (K) options operator, arrow; floor((log[10])(K))+1 end proc; A := proc (K) options operator, arrow; convert(K, base, 10) end proc; g := proc (K) options operator, arrow; sum(A(K)[i], i = 1 .. p(K)) end proc
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MATHEMATICA
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fQ[n_] := Block[{id = IntegerDigits@n}, Plus @@ id * Plus @@ (id^2) == n]; lst = {}; Do[ If[fQ@n, AppendTo[lst, n]], {n, 10^8}]; lst (* Robert G. Wilson v *)
For[n = 1, n < 30000, n++, b = IntegerDigits[n]; If[Sum[b[[i]], {i, 1, Length[b]}]*Sum[b[[i]]^2, {i, 1, Length[b]}] == n, Print[n]]] - Stefan Steinerberger
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CROSSREFS
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Sequence in context: A146233 A146181 A105984 * A064903 A070158 A055940
Adjacent sequences: A115515 A115516 A115517 * A115519 A115520 A115521
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KEYWORD
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base,fini,full,nonn
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AUTHOR
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Aktar Yalcin (aktaryalcin(AT)msn.com), Jun 24 2007
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EXTENSIONS
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Edited by Robert G. Wilson v and Stefan Steinerberger, Jun 28 2007
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STATUS
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approved
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