cp's OEIS Frontend

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

%I A186074 #69 Jan 29 2022 12:45:12
%S A186074 15,27,429,1353,1863,3388,3591,7119,78403,133533,178623,2282148,
%T A186074 2732353,3882813,7103835,13335333,17016076,17786223,27377889,32738728,
%U A186074 35639163,308725039,347826603,1248851513,1333353333,1420855168,1777862223,3146385338,3699393633
%N A186074 Numbers k such that k = Sum_{i=x..y} i and, in decimal, k is the concatenation of x and y.
%C A186074 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.
%C A186074 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.
%C A186074 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.
%C A186074 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.
%C A186074 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).
%C A186074 See A186076 for the case of a sum countdown from the more significant to less significant digits.
%C A186074 From _Jinyuan Wang_, Sep 14 2019: (Start)
%C A186074 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.
%C A186074 Strictly speaking, 1301613 = 13 + 14 + ... + 1612 + 1613 does not meet the concatenation criterion. So 1301613 is not a term.
%C A186074 (End)
%C A186074 From _Bernard Schott_, Jan 26 2022: (Start)
%C A186074 Note that numbers x are in A070152, and corresponding y in A070153 (see formula).
%C A186074 Other subsequence pattern: 27, 1863, 178623, 17786223, 1777862223, ... where 17..78 +...+ 62..23 = 17..7862..23. (End)
%H A186074 Jinyuan Wang, <a href="/A186074/b186074.txt">Table of n, a(n) for n = 1..18738</a> (first 142 terms from David A. Corneth)
%H A186074 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/1024-a1945-concatenations-en-tous-genres">A1945 - Concaténations en tous genres</a> (in French).
%H A186074 Richard Hoshino, <a href="http://cms.math.ca/crux/v27/n1/public_page34-47.pdf">Astonishing Pairs of Numbers</a>, Crux Mathematicorum with Mathematical Mayhem 27:1 (2001), pp. 39-44.
%F A186074 a(n) = A070152(n).A070153(n) where "." means concatenation. - _Bernard Schott_, Jan 29 2022
%e A186074 429 = 4 + 5 + 6 + ... + 28 + 29.
%e A186074 7119 = 7 + 8 + 9 + ... + 118 + 119.
%e A186074 3882813 = 388 + 389 + ... + 2812 + 2813.
%p A186074 # See "Astonishing Pairs of Numbers" article referenced above.
%o A186074 (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);
%o A186074 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
%Y A186074 Cf. A186076.
%Y A186074 Cf. A070152, A070153.
%Y A186074 Cf. A350994 (subsequence).
%K A186074 nonn,base
%O A186074 1,1
%A A186074 _Matthew Goers_, Feb 11 2011
%E A186074 a(22)-a(29) from _Matthew Goers_, Apr 11 2013