A186074 Numbers k such that k = Sum_{i=x..y} i and, in decimal, k is the concatenation of x and y.

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

Offset: 1

Author

Matthew Goers, Feb 11 2011

Keywords

• nonn
• base

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 15 = sum (1 to 5); 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 A350994).

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

From Jinyuan Wang, Sep 14 2019: (Start)

All terms form the concatenation of x = (s+t+1)/2 - 10^k and y = (s-t+1)/2, where s*t = 100^k - 10^k, 10^(k-1) < (s-t+1)/2 < 10^k, and gcd(s, t) is an odd number.

Strictly speaking, 1301613 = 13 + 14 + ... + 1612 + 1613 does not meet the concatenation criterion. So 1301613 is not a term.

(End)

From Bernard Schott, Jan 26 2022: (Start)

Note that numbers x are in A070152, and corresponding y in A070153 (see formula).

Other subsequence pattern: 27, 1863, 178623, 17786223, 1777862223, ... where 17..78 +...+ 62..23 = 17..7862..23. (End)

Examples

			429 = 4 + 5 + 6 + ... + 28 + 29.
7119 = 7 + 8 + 9 + ... + 118 + 119.
3882813 = 388 + 389 + ... + 2812 + 2813.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..18738 (first 142 terms from David A. Corneth)
• Diophante, A1945 - Concaténations en tous genres (in French).
• Richard Hoshino, Astonishing Pairs of Numbers, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.

Crossrefs

Cf. A186076.

Cf. A070152, A070153.

Cf. A350994 (subsequence).

Programs

Maple

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

PARI

do(s, t, k) = if(10^(k-1) < (s-t+1)/2 && (s-t+1)/2 < 10^k, (1-10^k+s)*(1+10^k-t)/2);
lista(nn) = {my(m, v=List()); for(k = 1, nn, fordiv(50^k - 5^k, s, t = (100^k-10^k)/s; if(m=do(s, t, k), listput(v, m)); if(m=do(2^k*s, t/2^k, k), listput(v, m)))); vecsort(Vec(v)); } \\ Jinyuan Wang, Aug 29 2019

Formula

a(n) = A070152(n).A070153(n) where "." means concatenation. - Bernard Schott, Jan 29 2022

Extensions

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