|
%I #26 Aug 07 2016 09:34:57
%S 11,605,4140,15464,320769,4499448,6569655,468939687,1800052998,
%T 76293876291,124999924997,8000003299997,39521606452157,
%U 146365371463650,2449999994499996,20000000169999996,3883989336388398929,40500000000049999995,3344565630038445656295,405000000000904999999995
%N Consider the function G(m) that adds to m a fractional part whose digits are the digits of m (informally, G(m) = m.m). Sequence lists integers of the form Sum_{i=1..k} G(i) for some k.
%C E.g.: G(54627) = 54627.54627.
%C Values of k for the terms here listed are: 4, 34, 90, 175, 800, 2999, 3624, 30624, 60000, 390624, 499999, 4000000, 8890624, 17109375, 69999999, 200000000, ... (see A054464).
%C (This sequence is in fact equivalent to A054464. - _N. J. A. Sloane_, Aug 07 2016)
%H Robert Israel, <a href="/A275572/b275572.txt">Table of n, a(n) for n = 1..900</a>
%F From _Robert Israel_, Aug 03 2016: (Start)
%F For d >=2, the k with d digits are the solutions of x^2 + x - 9*10^(d-1)*d - 10^(d-1) == 0 (mod 2*10^d) with 10^(d-1) <= x < 10^d.
%F The corresponding a(n) are k(k+1)(1+10^(-d))/2 + (10^d-9d-1)/20. (End)
%e 1.1 + 2.2 + 3.3 + 4.4 = 11;
%e 1.1 + 2.2 + 3.3 + ... + 32.32 + 33.33 + 34.34 = 605.
%p P:= proc(q) local a,b,c,k,n; c:=0; for n from 1 to q do a:=[]
%p b:=convert(n,base,10); for k from 1 to nops(b) do a:=[b[k],op(a)]; od;
%p a:=n+add(a[k]*10^(-k),k=1..nops(a));
%p c:=c+a; if type(c,integer) then print(c); fi; od; end: P(10^12);
%p # Alternative:
%p T := (x, d) -> ((1/2)*x^2+(1/2)*x)*10^(-d)+(1/2)*x^2-(9/20)*d+(1/2)*x+(1/20)*10^d-1/20:
%p F:= proc(d) local x,S;
%p S:= map(t -> subs(t, x), [msolve(x^2 + x - 9*10^(d-1)*d - 10^(d-1), 2*10^d)]);
%p op(map(T, sort(select(t -> t >= 10^(d-1) and t < 10^d, S)),d))
%p end proc:
%p 11, seq(F(d),d=2..30); # _Robert Israel_, Aug 03 2016
%t Select[Accumulate@ Map[# + #/10^IntegerLength@ # &, Range[10^7]], IntegerQ] (* _Michael De Vlieger_, Aug 02 2016 *)
%Y Cf. A054464, A275573.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Aug 02 2016
|
