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5-Smith numbers.
2

%I #16 Aug 23 2020 02:09:56

%S 2030,10203,12110,20210,20310,21004,21010,24000,24010,31010,41001,

%T 50010,70000,100004,100012,100210,100310,100320,101020,101041,102022,

%U 103200,104010,104101,104110,105020,106001,110020,110202,110212,110400,111013

%N 5-Smith numbers.

%H Amiram Eldar, <a href="/A103126/b103126.txt">Table of n, a(n) for n = 1..10000</a>

%H Shyam Sunder Gupta, <a href="http://www.shyamsundergupta.com/smith.htm">Smith Numbers</a>.

%H Wayne L. McDaniel, <a href="http://www.fq.math.ca/Scanned/25-1/mcdaniel.pdf">The Existence of infinitely Many k-Smith numbers</a>, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 76-80.

%e 2030 is a 5-Smith number because the sum of the digits of its prime factors, i.e., Sp(2030) = Sp(2*5*7*29) = 2 + 5 + 7 + 2 + 9 = 25, which is equal to 5 times the digit sum of 2030, i.e., 5*S(2030) = 5*(2 + 0 + 3 + 0) = 25.

%t digSum[n_] := Plus @@ IntegerDigits[n]; fiveSmithQ[n_] := CompositeQ[n] && Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) == 5 *digSum[n]; Select[Range[10^5], fiveSmithQ] (* _Amiram Eldar_, Aug 23 2020 *)

%Y Cf. A006753.

%K base,nonn

%O 1,1

%A _Shyam Sunder Gupta_, Mar 16 2005