A275572 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.

11, 605, 4140, 15464, 320769, 4499448, 6569655, 468939687, 1800052998, 76293876291, 124999924997, 8000003299997, 39521606452157, 146365371463650, 2449999994499996, 20000000169999996, 3883989336388398929, 40500000000049999995, 3344565630038445656295, 405000000000904999999995

Offset

1,1

Comments

E.g.: G(54627) = 54627.54627.

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).

(This sequence is in fact equivalent to A054464. - N. J. A. Sloane, Aug 07 2016)

Links

Robert Israel, Table of n, a(n) for n = 1..900

Formula

From Robert Israel, Aug 03 2016: (Start)

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.

The corresponding a(n) are k(k+1)(1+10^(-d))/2 + (10^d-9d-1)/20. (End)

Example

1.1 + 2.2 + 3.3 + 4.4 = 11;
1.1 + 2.2 + 3.3 + ... + 32.32 + 33.33 + 34.34 = 605.

Maple

P:= proc(q) local a, b, c, k, n; c:=0; for n from 1 to q do a:=[]
b:=convert(n, base, 10); for k from 1 to nops(b) do a:=[b[k], op(a)]; od;
a:=n+add(a[k]*10^(-k), k=1..nops(a));
c:=c+a; if type(c, integer) then print(c); fi; od; end: P(10^12);
# Alternative:
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:
F:= proc(d) local x, S;
  S:= map(t -> subs(t, x), [msolve(x^2 + x - 9*10^(d-1)*d - 10^(d-1), 2*10^d)]);
  op(map(T, sort(select(t -> t >= 10^(d-1) and t < 10^d, S)), d))
end proc:
11, seq(F(d), d=2..30); # Robert Israel, Aug 03 2016

Mathematica

Select[Accumulate@ Map[# + #/10^IntegerLength@ # &, Range[10^7]], IntegerQ] (* Michael De Vlieger, Aug 02 2016 *)

Crossrefs

Cf. A054464, A275573.

Sequence in context: A185656 A142738 A262015 * A115737 A319835 A288549

Adjacent sequences: A275569 A275570 A275571 * A275573 A275574 A275575

Keyword

nonn,base

Author

Paolo P. Lava, Aug 02 2016

Status

approved