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A186074 Numbers n such that n = sum_{k=i..j} k and, in decimal, n is the concatenation of i and j. 2
15, 27, 429, 1353, 1863, 3388, 3591, 7119, 78403, 133533, 178623, 2282148, 2732353, 3882813, 7103835, 13335333, 17016076, 17786223, 27377889, 32738728, 35639163, 308725039, 347826603, 1248851513, 1333353333, 1420855168, 1777862223, 3146385338, 3699393633 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The sum from one set of digits to the following set of digits equals the term.  The first is the 5th triangular number: 15 = 1 + 2 + 3 + 4 + 5.

These are the positive integer solutions for the formula sum (x to y) = (10^k)*x + y, where 0 < x < y < 10^k for some k>=1.

On the left hand side of this equation, the sum can be written as A000217(y) - A000217(x-1) = (x+y)*(1-x+y)/2, and the right hand side is the concatenation of the decimal digits of x and y.

The graph of the function is a hyperbola; the solutions are for positive x and y, where y does not "overlap" and add to x.  The first 21 terms are all of the solutions for n = 1 to 4. n = 5 solutions add two 9-digit and six 10-digit terms.

Note the pattern 1353 = sum (13 to 53); 133533 = sum (133 to 533); 13335333 = sum (1333 to 5333). This pattern continues: 1333353333 = sum (13333 to 53333); 133333533333 = sum (133333 to 533333); etc.  These are not the next terms in the sequence, however.

See A186076 for the case of a sum countdown from the more significant to less significant digits.

LINKS

Matthew Goers, Table of n, a(n) for n = 1..29

Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.

EXAMPLE

429 = 4 + 5 + 6 + ... + 28 + 29.

7119 = 7 + 8 + 9 + ... + 118 + 119.

3882813 = 388 + 389 + ... + 2812 + 2813.

MAPLE

# See "Astonishing Pairs of Numbers" article referenced above.

CROSSREFS

Cf. A186076.

Sequence in context: A087719 A174216 A116070 * A230649 A229195 A073766

Adjacent sequences:  A186071 A186072 A186073 * A186075 A186076 A186077

KEYWORD

nonn,base

AUTHOR

Matthew Goers, Feb 11 2011

EXTENSIONS

Added term a(22) - a(29), Matthew Goers, Apr 11 2013

STATUS

approved

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Last modified June 25 20:26 EDT 2017. Contains 288730 sequences.