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# User talk:Ralf Steiner

The shortest chain is: 2, 3, 5, 8, 11, 17, 24, 29, 39, 48, 55, 65, 77, 89, 103, 119, 128, 143, 156, 171, 190, 208, 227, 245, 264, ... Ralf Steiner (talk) 11:27, 17 February 2019 (EST)

The longest chain is: 2, 3, 4, 6, 8, 10, 15, 18, 24, 28, 35, 42, 49, 56, 66, 79, 95, 105, 120, 128, 143, 156, 171, 190, 208, 225, 244, 264, ... Conjecture: In a n-sequence of square submatrices T_n, n > 0 of A082786 adjusted for blank columns with exactly one overlapping row a(n) only, the closed interval of rows is [a(n), a(n + 1)]. An only main chain [a(1),a(2)]-[a(2),a(3)]-[a(3),a(4)]-... exists with chain links [,] of minimum lenght each - this sequence - including one or two triangular numbers (A000217) each and all determinants of the related submatrices T_n are not equal to 0. Ralf Steiner (talk) 05:29, 17 February 2019 (EST)

All possible (not only the longest) chain links -[,]- with their determinants in () build a directed graph structure of an inverse tree as shown below (limited on numbers <= 10) with the infinite main chain on top and finite secondary chains below reconnected with chains above by '+'.

2: [2(1)3]-[3(-2)5]-[5(3)8]-[8(6)10]-... | +3: +-+-+-... [3(-2)4]-[4(-2)7]-[7(-6)10]-... | +4: +-+-... [4(-2)6]-[6(3)8]-+-... [4(2)5]-+-+-... | +5: +-+-... | +6: +-+-... | +7: +-... [7(6)9]-... [7(-3)8]-+-... | +8: +-... [8(6)9]-... | +9:

Ralf Steiner (talk) 08:28, 15 February 2019 (EST)

There an only main chain [,]-[,]-[,]-... exists with chain links consisting of minimal overlapping closed intervals [,] of maximum lenght each: {2, 3, 5, 8, 11, 17, 24, 29, 39, 48, 55, 65, 77, 89, 103, 119, 131, 149, 167, 186, ...} including exactly one triangular number each, such as [2,3]-[3,5]-[5,8]-[8,11]- include the 3, 6 or 10 respectively.

Ralf Steiner (talk) 02:57, 13 February 2019 (EST)

Dear number theorists,

My answers to your questions are:

1) With calculation of the determinants and check if it is not null each. With try and error we get a tree-like structure of linked intervals. The common head of all these possible paths in this tree is a singular sequence for both the intervals and for the determinants: {2, 3, 4, 6, 8, 10, 15, 18, 24, 28, 35, 42, 49, 56, 66, 77, 88, ... }; {1, -2, -2, 3, 6, 3, -8, -5, -18, -5, -6, 6, -36, 12, 5, -20, ... }. I will explain it more in detail when these new sequences are filed in OEIS.

2) The triangular numbers T(n) are a necessary part of the conjecture. Further there are (well-known, but not yet proved) relation between the primes and the triangular number interval, such as each interval [T(n - 1) + 1,T(n)], n > 1 contains at least one prime p (see A065382).

Ralf Steiner (talk) 11:35, 31 January 2019 (EST)

"Dear number theorists,

I would like to inform you about a further relation between primes and triangular numbers T(n) (A000217 <https://oeis.org/A000217> in OEIS) and maybe discuss this a bit. Note for this new relation please: https://oeis.org/A082786

My conjecture is: All determinants of (in a sequence overlapping) square submatrices adjusted for blank columns with the rows in at least one closed interval including at least one triangular number A000217 are not equal to 0. With exactly one overlapping row only this interval is [n(j-1), n(j)].

With n(j)={2, 3, 4, 6, 8, 10, 15, 18, 24, 27, 32, 40, 48, 55, 64, 75, 84, 98, 108} we get for the determinants: {1, -2, -2, 3, 6, 3, -8, -5, 18, -15, -10, 11, 44, -12, -24, -32, 7, -26}.

For intervals including triangular numbers take for example: {2, 3, 4, 6, 8, 10, 15, 18, 24, 28, 35, 42, 49, 56, 66, 77, 89, 100, 112} => {1, -2, -2, 3, 6, 3, -8, -5, -18, -5, -6, 6, -36, 12, 5, -20, -40, 34}. Or take {2, 3, 4, 6, 8, 10, 15, 18, 24, 28, 35, 42, 49, 56, 66, 77, 88, 99, 110, 125, 136} => {1, -2, -2, 3, 6, 3, -8, -5, -18, -5, -6, 6, -36, 12, 5, -20, 26, 8, 6, -63}. Note the common head of the det-sequences.

=> Thus there is a very interesting sequence related to the fundamental theorem of arithmetic: {1, -2, -2, 3, 6, 3, -8, -5, -18, -5, -6, 6, -36, 12, 5, -20, ... }. ==>> This sequence should be inserted to the OEIS (I don't have a free slot recently for this)."

1) Hi, how did you arrive to the suggested by you sequence {1, -2, -2, 3, 6, 3, -8, -5, -18, -5, -6, 6, -36, 12, 5, -20, ... } from two previous examples, which you gave above it ? 2) Where do you see the relation between primes and triangular numbers in the suggested by you sequence?

Cheers, A.P.