

A186076


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.


2



190, 204, 216, 19900, 20328, 21252, 21762, 23287, 23490, 1999000, 2002077, 2006118, 2077402, 2132532, 2177622, 199990000, 202272147, 202722352, 203872812, 207093834, 213325332, 217006075, 217776222, 227367888, 232728727, 235629162, 19999900000, 20001201612
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OFFSET

1,1


COMMENTS

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.
See A186074 for numbers that sum by counting upward.
An alternate definition: n = sum from x to y inclusive = A000217(y)  A000217(x1), (A000217 are the triangular numbers) where the digits of n are the concatenation of y and x.
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...
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.
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
All terms form (10^k)*y + x, where y = (s+t1)/2 + 10^k, x = (st1)/2, s*t = 100^k  10^k, 0 <= (st1)/2 < 10^k, and gcd(s, t) is an odd number.  Jinyuan Wang, Sep 13 2019


LINKS

Jinyuan Wang, Table of n, a(n) for n = 1..21167


EXAMPLE

204 = 20 + 19 + 18 + ... + 5 + 4.
2002077 = 2002 + 2001 + ... + 78 + 77.
2006118 = 2006 + 2005 + ... + 119 + 118.


PROG

(PARI) do(s, t, k) = if(s  t > 0 && (st1)/2 < 10^k, (10^k1+s)*(10^k+1+t)/2, 204);
lista(nn) = {my(v=List()); for(k = 1, nn, fordiv(50^k  5^k, s, t = (100^k10^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


CROSSREFS

Cf. A186074, A000217.
Sequence in context: A025391 A025382 A179518 * A260762 A218292 A333834
Adjacent sequences: A186073 A186074 A186075 * A186077 A186078 A186079


KEYWORD

nonn,base


AUTHOR

Matthew Goers, Feb 11 2011


EXTENSIONS

Missing term a(4) = 19900 inserted by Matthew Goers, Jun 03 2011
a(16)a(28) from Donovan Johnson, Aug 22 2012


STATUS

approved



