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.

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

Offset: 1

Author

Matthew Goers, Feb 11 2011

Keywords

• nonn
• base

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(x-1), (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+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

Examples

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

Links

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

Crossrefs

Cf. A186074, A000217.

Programs

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);
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

Extensions

Missing term a(4) = 19900 inserted by Matthew Goers, Jun 03 2011

a(16)-a(28) from Donovan Johnson, Aug 22 2012