A103123 - OEIS

%I #19 Jan 05 2025 19:51:38

%S 19899699,36969999,36999699,39699969,39999399,39999993,66699699,

%T 66798798,67967799,67987986,69759897,69889389,69966699,69996993,

%U 76668999,79488798,79866798,85994799,86686886,89769759,89866568

%N 1/4-Smith numbers.

%H Amiram Eldar, <a href="/A103123/b103123.txt">Table of n, a(n) for n = 1..1000</a>

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

%H Wayne L. McDaniel, <a href="https://web.archive.org/web/2024*/https://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 19899699 is a 4^(-1) Smith number because the digit sum of 19899699, i.e., S(19899699) = 1 + 9 + 8 + 9 + 9 + 6 + 9 + 9 = 60, which is equal to 4 times the sum of the digits of its prime factors, i.e., 4*Sp(19899699) = 4*Sp (3*2203*3011) = 4*(3 + 2 + 2 + 0 + 3 + 3 + 0 + 1 + 1) = 15.

%t digSum[n_] := Plus @@ IntegerDigits[n]; qSmithQ[n_] := CompositeQ[n] && 4 * Plus @@ (Last@# * digSum[First@#] & /@ FactorInteger[n]) ==  digSum[n]; Select[Range[10^8], qSmithQ] (* _Amiram Eldar_, Aug 23 2020 *)

%Y Cf. A006753.

%K base,nonn

%O 1,1

%A _Shyam Sunder Gupta_, Mar 16 2005