new sequences

[Andrew Plewe]

This is similar to a sequence I submitted for squares:

http://www.research.att.com/~njas/sequences/A129861

I think it's worth noting that a.) this can be expanded to all figurate 
numbers, and b.) that any figurate number works in Fermat's factoring 
method, i.e. if F_n = nth figurate number, then finding a nontrivial 
solution to:

F_n(a) - F_n(b) = x

will reveal a factor of x (by taking GCD(x, a-b)). Not all numbers have 
valid representations for all F_n. The number of representation depends 
on the ratio of the factors. So, for instance, RSA numbers have 
solutions for F_1 and F_2 (the triangle numbers and squares, 
respectively), and some may go as high as F_4 or F_5, but there are no 
solutions for "higher" figurate numbers (generally speaking). I have on 
my to-do list generating two sequences based on figurate numbers. One is 
a sequence of the smallest number a such that F_n(a) - F_n(b) = x over 
all values of n. The other is the smallest value for n such that the 
same thing is true.

     -Andrew Plewe-


> %I A138797
> %S A138797 3,6,10,6,21,10,36,10,55,21,15,28,15,21,136,45,21,55,21,36,28,78,45,28,
> %T A138797 36,28,406,120,36,136,528,36,55,36,91,190,66,45,55,231,45,253,45,55,91,
> %U A138797 300,153,55,78,66,55,378,55,91,66,78,136,465,66,496,153,66,2080,66,171
> %N A138797 Least possible T(k) with T(k)-T(j)=n, where T(i)>0 are the
> triangular numbers A000217
> %Y A138797 Cf. A000217,A109814,A118235,A136107,A138796,A138798,A138799.
> %O A138797 2,1
> %e A138797 a(30)=8, because 30 = T(30)-T(29)=T(11)-T(8)=T(9)-T(5)=T(8)-T(3)
> and 8 is the least index of the minuends
> %t A138797
> T=#(#+1)/2&;T[Min[k/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0<j<k},{j,k},Integers]]}]]&/@Range[2,100]
> %K A138797 nonn
> %C A138797 for k see A138796, for j see A138798 and for T(j) see A138799.
> %C A138797 The number of ways n can be written as difference of two triangular
> numbers is sequence A136107
> %A A138797 Peter Pein (petsie(AT)dordos.net), Mar 30, 2008