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%I #23 May 11 2023 10:26:46
%S 0,19,1047,64373,4481640,340900331,27436000061,2292176360707,
%T 196818634871899,17246903703574357,1534951275195670059,
%U 138293592048140425181,12583738258227621100170,1154435823206834353336284,106638384745041347295504523
%N Sum of the proper prime divisors of all numbers up to 10^n.
%C The sum of the distinct prime factors less than k for all 1 <= k <= 10^n, as tabulated for the individual k in A105221.
%F a(n) = Sum_{k=1..10^n} A105221(k). - _R. J. Mathar_, Dec 12 2007
%F a(n) = Sum_{prime p<10^n} p*floor((10^n-p)/p) = A006880(n)*10^n - A024934(10^n) - A046731(n). - _Max Alekseyev_, Jan 30 2012
%e a(1)=19 because 10^1=10 and the factors to be summed are 2 for 4, added to 2 and 3 for 6, added to 2 for 8, added to 3 for 9, added to 2 and 5 for 10.
%p A105221 := proc(n) local a,pfs,i ; a :=0 ; pfs := ifactors(n)[2] ; for i in pfs do if op(1,i) <> 1 and op(1,i) <> n then a := a+op(1,i) ; fi ; od: RETURN(a) ; end: A136021 := proc(n) add(A105221(i),i=2..10^n) ; end: for n from 1 do print(n,A136021(n)) ; od: # _R. J. Mathar_, Dec 12 2007
%t f[n_] := Plus @@ (First@# & /@ FactorInteger@ n); k = 2; s = 0; lst = {}; Do[While[k < 10^n + 1, If[ ! PrimeQ@k, s = s + f@k]; k++ ]; AppendTo[ lst, s]; Print[{n, s}], {n, 8}] (* _Robert G. Wilson v_, Aug 06 2010 *)
%o (UBASIC) 10 'distinct prime factors of composites <=10^n 20 S=0:N=N+1:Z=N\2 30 'print N; 40 for F=1 to Z:Q=N/F: if Q<>int(Q) then 60 50 S=S+F: if F=prmdiv(F) and F>1 then C=C+1:G=G+F 60 next F 70 'print C,G 80 if N=10^1 or N=10^2 or N=10^3 or N=10^4 or N=10^5 or N=10^6 or N=10^7 then print G:stop 90 C=0 100 goto 20
%Y Cf. A136022, A136023, A136024, A136025.
%K more,nonn
%O 0,2
%A _Enoch Haga_, Dec 10 2007
%E One more term from _R. J. Mathar_, Dec 12 2007
%E Edited by _R. J. Mathar_, Apr 17 2009
%E a(7) & a(8) from _Robert G. Wilson v_, Aug 06 2010
%E a(9)-a(11) from _Max Alekseyev_, Jan 30 2012
%E a(12)-a(14) from _Hiroaki Yamanouchi_, Jun 29 2014
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