Sequence A067270: Nth Triangular Number Ends in N
This sequence, Sloane's A67270, contains all numbers n such that the nth triangular number is equal to n modulo 10d, where d is the number of digits in n, i.e. d=floor(log10(n)+1). Sequence A67270 is: 0, 1, 5, 25, 625, 9376, 90625, 890625, 7109376, 12890625, 212890625, 1787109376, 81787109376, 59918212890625, 259918212890625, 3740081787109376, 56259918212890625, 256259918212890625, 7743740081787109376, ...
The OEIS entry for A67270 states,
Thanks to David W. Wilson for the proof that this sequence is a proper subset of A3226.
A3226 are the "square automorphic" numbers, normally called simply "automorphic". Curiously, there are no relevant links, nor are there any hints or clues in A3226 or in related sequences such as A7185, A18247, A18248, A16090, etc. However, a proof is easy to work out.
Proof that A67270 implies A3226
Let T(n) denote the nth triangular number, n(n+1)/2. As stated above, we also let d be the number of digits in n, d=floor(log10(n)+1).
Given: T(n)≡n mod 10d, we wish to prove that n2≡n mod 10d.
From T(n), subtract n to get n(n+1)/2-n = n((n+1)/2-1) = n((n+1)/2-2/2) = n((n+1-2)/2) = n(n-1)/2.
Because T(n)≡n mod 10d, and n≡n mod 10d, we know that (T(n)-n)≡0 mod 10d.
Therefore, (T(n)-n)+T(n) must be ≡n mod 10d.
But (T(n)-n)+T(n) = n(n-1)/2 + n(n+1)/2 = n(n-1+n+1)/2 = n(2n)/2 = n×n = n2.
Therefore, n2≡n mod 10d.
Examples
We can illustrate this with a table of examples:
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It's retty easy to see from these examples: the numbers in the 1st column are the triangular automorph numbers, so the corresponding triangular numbers in the 2nd column end with the digits of n. Therefore, when you subtract column 1 from column 2 you get something ending in 0's, and then when adding column 2 to column 3 you get something that again ends with the digits of n.
A67270 is a Proper Subset of A3226
We have shown that if a number is in A67270, then it is also in A3226. However, the opposite is not true: there are additional numbers in A3226 (such as 376) that are "square automorphic" without being triangular automorphic.
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