A275573 Consider the function G(n) that adds to n a decimal part equal to the reverse of n. Sequence lists integers that are equal to Sum_{i=1..k}{G(i)} for some k.

11, 1386327162, 1644167162, 98457031244, 138632716794, 164416716794, 215332031244, 3164184570305, 9845703124994, 13863271679994, 16441671679994, 21533203124994, 35992309570305, 984570312499994, 1386327167999994, 1644167167999994, 2153320312499994, 80783157348632805, 98457031249999994, 138632716799999994, 164416716799999994, 215332031249999994, 243634719848632805

Offset

1,1

Comments

E.g.: G(54627) = 54627.72645

All terms are multiples of 11.

Values of k for the terms here listed are 4, 52655, 57343, 443749, 526559, 573439, 656249, 2515624, 4437499, 5265599, 5734399, 6562499, 8484374, 44374999, 52655999, 57343999, 65624999, ...

The sequence is infinite. Since Sum_{i=1..57343}G(i) and Sum_{i=57344..573439}G(i) are both integers, this implies that Sum_{i=1..57344*10^j - 1}G(i) is a term for j >= 0. In particular, if Sum_{i=1..a-1}G(i) and Sum_{i=1..10a-1}G(i) are both terms of the sequence and a is even, then Sum_{i=1..a*10^j-1}G(i) are terms for j >= 0 and Sum_{i=a..10a-1}(G(i)-i) + 81*a/2 = Sum_{i=10a..100a-1}(G(i)-i). Thus Sum_{i=1..52656*10^j - 1}G(i), Sum_{i=1..443750*10^j - 1}G(i) and Sum_{i=1..656250*10^j - 1}G(i) are also terms for j >= 0. - Chai Wah Wu, Aug 24 2016

Links

Table of n, a(n) for n=1..23.

Example

1.1 + 2.2 + 3.3 + 4.4 = 11;
1.1 + 2.2 + ... + 52654.45625 + 52655.55625 = 1386327162.

Maple

P:= proc(q) local a, b, k, n; b:=0; for n from 1 to q do
a:=convert(n, base, 10); a:=n+add(a[k]*10^(-k), k=1..nops(a));
b:=b+a; if type(b, integer) then print(b); fi; od; end: P(10^12);

Mathematica

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

Prog

(Python)
from __future__ import division
from fractions import Fraction
A275573_list, q = [], 0
for i in range(1, 10**6):
    q += Fraction(int(str(i)[::-1]), 10**len(str(i)))
    if q.denominator == 1:
        A275573_list.append(q + i*(i+1)//2) # Chai Wah Wu, Aug 24 2016

Crossrefs

Cf. A275572.

Sequence in context: A022545 A346996 A086503 * A247846 A257127 A027569

Adjacent sequences: A275570 A275571 A275572 * A275574 A275575 A275576

Keyword

nonn,base

Author

Paolo P. Lava, Aug 02 2016

Extensions

a(18)-a(23) from Chai Wah Wu, Aug 24 2016

Status

approved