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%I #25 Aug 25 2016 09:26:26
%S 11,1386327162,1644167162,98457031244,138632716794,164416716794,
%T 215332031244,3164184570305,9845703124994,13863271679994,
%U 16441671679994,21533203124994,35992309570305,984570312499994,1386327167999994,1644167167999994,2153320312499994,80783157348632805,98457031249999994,138632716799999994,164416716799999994,215332031249999994,243634719848632805
%N 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.
%C E.g.: G(54627) = 54627.72645
%C All terms are multiples of 11.
%C 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, ...
%C 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
%e 1.1 + 2.2 + 3.3 + 4.4 = 11;
%e 1.1 + 2.2 + ... + 52654.45625 + 52655.55625 = 1386327162.
%p P:= proc(q) local a,b,k,n; b:=0; for n from 1 to q do
%p a:=convert(n,base,10); a:=n+add(a[k]*10^(-k),k=1..nops(a));
%p b:=b+a; if type(b,integer) then print(b); fi; od; end: P(10^12);
%t Select[Accumulate@ Map[# + FromDigits[Reverse@ IntegerDigits@ #]/10^IntegerLength@ # &, Range[10^7]], IntegerQ] (* _Michael De Vlieger_, Aug 02 2016 *)
%o (Python)
%o from __future__ import division
%o from fractions import Fraction
%o A275573_list, q = [], 0
%o for i in range(1,10**6):
%o q += Fraction(int(str(i)[::-1]),10**len(str(i)))
%o if q.denominator == 1:
%o A275573_list.append(q + i*(i+1)//2) # _Chai Wah Wu_, Aug 24 2016
%Y Cf. A275572.
%K nonn,base
%O 1,1
%A _Paolo P. Lava_, Aug 02 2016
%E a(18)-a(23) from _Chai Wah Wu_, Aug 24 2016
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