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%I #13 Dec 31 2021 16:09:01
%S 4224,6235,82226,9336,10257,122237,14279,15358,1622228,182338,202259,
%T 213710,2221113,2422239,255510,2621315,273339,2822711,3023510,
%U 322222210,3331114,3421719,355712,36223310,3821921,3931316
%N Concatenate n-th composite integer with concatenation of its prime factors in ascending order and the sum of its prime factors.
%H Jens Kruse Andersen, <a href="/A245316/b245316.txt">Table of n, a(n) for n = 1..10000</a>
%e a(2)=6235 since 6=2*3, 2+3=5 and 6 is the second composite integer.
%t f[n_]:=FactorInteger[n];con[A_]:=(v1={};l=Length[A];Do[v1=Join[v1,IntegerDigits[A[[k]]]],{k,l}];FromDigits[v1]);alfa[n_]:=(b=f[n];j=Length[b];c=Table[Table[b[[k]][[1]],{b[[k]][[2]]}],{k,j}];w={};Do[w=Join[w,c[[k]]],{k,j}];con[w]);omega[n_]:=(b=f[n];j=Length[b];c=Table[Table[b[[k]][[1]],{b[[k]][[2]]}],{k,j}];w={};Do[w=Join[w,c[[k]]],{k,j}];Total[w]);nao[n_]:=con[{n,alfa[n],omega[n]}];v=Select[Range[2,1000],!PrimeQ[#]&];Table[nao[v[[k]]],{k,26}]
%t compcat[n_]:=Module[{f=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[ n]]},FromDigits[ Join[IntegerDigits[n],Flatten[ IntegerDigits/@ f],IntegerDigits[ Total[f]]]]]; compcat/@Select[Range[40],CompositeQ] (* _Harvey P. Dale_, Dec 31 2021 *)
%Y Cf. A002808, A245315.
%K nonn,base
%O 1,1
%A _Jahangeer Kholdi_, Sep 15 2014