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A136021 Sum of the proper prime divisors of all numbers up to 10^n. 5
0, 19, 1047, 64373, 4481640, 340900331, 27436000061, 2292176360707, 196818634871899, 17246903703574357, 1534951275195670059, 138293592048140425181, 12583738258227621100170, 1154435823206834353336284, 106638384745041347295504523 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The sum of the distinct prime factors less than k for all 1 <= k <= 10^n, as tabulated for the individual k in A105221.

LINKS

Table of n, a(n) for n=0..14.

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

EXAMPLE

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.

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

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 *)

PROG

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

CROSSREFS

Cf. A136022, A136023, A136024, A136025.

Sequence in context: A285862 A136022 A203582 * A195214 A213864 A217827

Adjacent sequences:  A136018 A136019 A136020 * A136022 A136023 A136024

KEYWORD

more,nonn

AUTHOR

Enoch Haga, Dec 10 2007

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

STATUS

approved

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Last modified April 17 16:59 EDT 2021. Contains 343063 sequences. (Running on oeis4.)