A136021 Sum of the proper prime divisors of all numbers up to 10^n.
0, 19, 1047, 64373, 4481640, 340900331, 27436000061, 2292176360707, 196818634871899, 17246903703574357, 1534951275195670059, 138293592048140425181, 12583738258227621100170, 1154435823206834353336284, 106638384745041347295504523
Offset: 0
Examples
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.
Programs
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Maple
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
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Mathematica
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 *) -
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
Formula
a(n) = Sum_{k=1..10^n} A105221(k). - R. J. Mathar, Dec 12 2007
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
Extensions
One more term from R. J. Mathar, Dec 12 2007
Edited by R. J. Mathar, Apr 17 2009
a(7) & a(8) from Robert G. Wilson v, Aug 06 2010
a(9)-a(11) from Max Alekseyev, Jan 30 2012
a(12)-a(14) from Hiroaki Yamanouchi, Jun 29 2014
Comments