A186076 - OEIS

%I #32 Sep 24 2019 09:36:36

%S 190,204,216,19900,20328,21252,21762,23287,23490,1999000,2002077,

%T 2006118,2077402,2132532,2177622,199990000,202272147,202722352,

%U 203872812,207093834,213325332,217006075,217776222,227367888,232728727,235629162,19999900000,20001201612

%N Numbers m such that m = Sum_{i=x..y} i = (10^k)*y + x, where 0 <= x < y, 0 <= x < 10^k for some positive integers k.

%C Numbers that are the sum from their more significant digits counted down to the following digits. The first is the 19th triangular number: 190 = 19 + 18 + 17 + ... + 1 + 0.

%C See A186074 for numbers that sum by counting upward.

%C An alternate definition: n = sum from x to y inclusive = A000217(y) - A000217(x-1), (A000217 are the triangular numbers) where the digits of n are the concatenation of y and x.

%C These are the positive integer solutions to the equation Sum_{i=x..y} i = (10^k)*y + x, where 0 <= x < y, 0 <= x < 10^k, k = 1,2,3...

%C The graph of the function is a hyperbola; the solutions are for positive x and y, where x does not "overlap" and add to y. The first 15 terms are all of the solutions for m = 1 to 3.

%C Note that terms A186074(4) and A186074(10) have trailing 0's, i.e. 19900 = Sum_{k=0..199} k and 1999000 = Sum_{k=0..1999} k. Strictly speaking, these do not meet the concatenation criterion. This pattern continues indefinitely: 199990000, 19999900000, etc. - _Matthew Goers_, Jun 03 2011

%C All terms form (10^k)*y + x, where y = (s+t-1)/2 + 10^k, x = (s-t-1)/2, s*t = 100^k - 10^k, 0 <= (s-t-1)/2 < 10^k, and gcd(s, t) is an odd number. - _Jinyuan Wang_, Sep 13 2019

%H Jinyuan Wang, <a href="/A186076/b186076.txt">Table of n, a(n) for n = 1..21167</a>

%e 204 = 20 + 19 + 18 + ... + 5 + 4.

%e 2002077 = 2002 + 2001 + ... + 78 + 77.

%e 2006118 = 2006 + 2005 + ... + 119 + 118.

%o (PARI) do(s, t, k) = if(s - t > 0 && (s-t-1)/2 < 10^k, (10^k-1+s)*(10^k+1+t)/2, 204);

%o lista(nn) = {my(v=List()); for(k = 1, nn, fordiv(50^k - 5^k, s, t = (100^k-10^k)/s; listput(v, do(s, t, k)); listput(v, do(2^k*s, t/2^k, k)))); Set(v); } \\ _Jinyuan Wang_, Sep 13 2019

%Y Cf. A186074, A000217.

%K nonn,base

%O 1,1

%A _Matthew Goers_, Feb 11 2011

%E Missing term a(4) = 19900 inserted by _Matthew Goers_, Jun 03 2011

%E a(16)-a(28) from _Donovan Johnson_, Aug 22 2012