cp's OEIS Frontend

Square table T, read by ascending antidiagonals, where T(n,m) gives the least integer in the n-th occurrence of a run of exactly m consecutive integers in the ordered sequence A270877.

A270877 is sifted from the positive integers by modifying the sieve of Eratosthenes: instead of eliminating integers that would enumerate a rectangular area dot pattern with one side held at a constant length (equal to each surviving integer in turn), the sieve eliminates those enumerating a trapezoidal area dot pattern with the constant length being the trapezoid's longest side. Given this geometric relationship, it is considered worth looking for qualities that A270877 may have in common with the sequence of primes, potentially influenced by related causes such as the effect of prime factors on A270877.

The columns of this sequence, listing the runs of m consecutive integers within A270877, merit comparative examination with equivalent sequences for prime k-tuples. For m=5, the notably larger ratio between T(1,5) and T(2,5) resembles early large ratio gaps in the occurrence sequences of k-tuples such as A022008 (sextuples), whereas columns m<5 are more comparable with those for shorter k-tuples such as A001359 (twin primes) and A007530 (quadruples), each having a relatively low-valued first term (less than 60) and without such a large ratio gap. In comparison, the columns for runs m>5 appear more like the sequences for some longer k-tuples such as A027570 (a 10-tuple sequence). Row 1 merits comparative examination with A186702 for primes.

The author conjectures that T(n,m) exists for all n>=1, m>=1.

As a rectangle, the accumulation array of A051340.

From Clark Kimberling, Feb 05 2011: (Start)

Here all the weights are divided by two where they aren't in Cahn.

As a rectangle, A141419 is in the accumulation chain

... < A051340 < A141419 < A185874 < A185875 < A185876 < ...

(See A144112 for the definition of accumulation array.)

row 1: A000027

col 1: A000217

diag (1,5,...): A000326 (pentagonal numbers)

diag (2,7,...): A005449 (second pentagonal numbers)

diag (3,9,...): A045943 (triangular matchstick numbers)

diag (4,11,...): A115067

diag (5,13,...): A140090

diag (6,15,...): A140091

diag (7,17,...): A059845

diag (8,19,...): A140672

(End)

Let N=2*n+1 and k=1,2,...,n. Let A_{N,n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unit-primitive matrix (see [Jeffery]). Let M_n=[A_{N,n-1}]^4. Then t(n,k)=[M_n](1,k), that is, the n-th row of the triangle is given by the first row of M_n. - _L. Edson Jeffery, Nov 20 2011

Conjecture. Let N=2*n+1 and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (again see [Jeffery]) associated with N, and define the Chebyshev polynomials of the second kind by the recurrence U_0(x) = 1, U_1(x) = 2*x and U_r(x) = 2*x*U_(r-1)(x) - U_(r-2)(x) (r>1). Define the column vectors V_(k-1) = (U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where T denotes matrix transpose. Let S_N = [V_0, V_1, ..., V_(n-1)] be the n X n matrix formed by taking V_(k-1) as column k-1. Let X_N = [S_N]^T*S_N, and let [X_N](i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then t(n,k) = [X_N](k-1,k-1), and row n of the triangle is given by the main diagonal entries of X_N. Remarks: Hence t(n,k) is the sum of squares t(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N = [sum_{m=1,...,n} A057059(n,m)*A_{N,m-1}] is also an integral linear combination of unit-primitive matrices from the N-th set. - L. Edson Jeffery, Jan 20 2012

Row sums: n*(n+1)*(2*n+1)/6. - L. Edson Jeffery, Jan 25 2013

n-th row = partial sums of n-th row of A004736. - Reinhard Zumkeller, Aug 04 2014

T(n,k) is the number of distinct sums made by at most k elements in {1, 2, ... n}, for 1 <= k <= n, e.g., T(6,2) = the number of distinct sums made by at most 2 elements in {1,2,3,4,5,6}. The sums range from 1, to 5+6=11. So there are 11 distinct sums. - Derek Orr, Nov 26 2014

A number n occurs in this sequence A001227(n) times, the number of odd divisors of n, see A209260. - Hartmut F. W. Hoft, Apr 14 2016

Conjecture: 2*n + 1 is composite if and only if gcd(t(n,m),m) != 1, for some m. - L. Edson Jeffery, Jan 30 2018

From Peter Munn, Aug 21 2019 in respect of the sequence read as a triangle: (Start)

A number m can be found in column k if and only if A286013(m, k) is nonzero, in which case m occurs in column k on row A286013(m, k).

The first occurrence of m is in row A212652(m) column A109814(m), which is the rightmost column in which m occurs. This occurrence determines where m appears in A209260. The last occurrence of m is in row m column 1.

Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A270877; this is the only one that contains its own row numbers only once.

(End)

Conjecture 1: T(n,k) is the largest part of the partition of n into k consecutive parts, if T(n,k) > 0.

Conjecture 2: row sums give A286015.

Trapezoidal interpretation from Peter Munn, Jun 18 2017: (Start)

There is one to one correspondence between nonzero T(n,k) and trapezoidal area patterns of n dots on a triangular grid, if we include the limiting cases of triangular patterns, straight lines (k=1) or a single dot (k=n=1). The corresponding pattern has T(n,k) dots in its longest side, k dots in the two adjacent sides and T(n,k)-k+1 dots in the fourth side (where a count of 1 dot may be understood as signifying that side's absence).

Reason: From the definition, for k >= 1, m >= 0, T(A000217(k)+km,k) = k+m, where A000217(k) = k(k+1)/2, the k-th triangular number. First element of column k is T(A000217(k),k) = k: this matches a triangular pattern of A000217(k) dots with 3 sides of k dots. Looking at this pattern as k rows of 1..k dots, extend each row by m dots to create a trapezoidal pattern of A000217(k)+km dots with a longest side of k+m dots and adjacent sides of k dots: this matches T(A000217(k)+km,k) = k+m. As nonzero elements in column k occur at intervals of k, every nonzero T(n,k) has a match. Every trapezoidal pattern can be produced by extending a triangular pattern as described, so they all have a match.

The truth of conjecture 1 follows, since each nonzero T(n,k) = k+m corresponds to a trapezoidal pattern of n dots having k rows with lengths (1+m)..(k+m).

The A270877 sieve is related to this sequence because it eliminates n if it is the sum of consecutive numbers whose largest term has survived the sifting (which may likewise be seen in terms of a trapezoidal dot pattern and its longest side). So the sieve eliminates n if any lesser numbers in A270877 are in row n of this sequence.

(End)

A099104(a(n)) = 1.

a(A207432(n)) = A000040(n). - Reinhard Zumkeller, Feb 17 2012

Obviously all primes and cubes of primes are in the sequence, while squares of primes are not. In fact, A000225 tells us which exponents prime powers in the sequence will exhibit.

But where it gets really interesting is in what happens to the Achilles numbers: the smallest badly sieved numbers that are also Achilles numbers are 864 and 972. - Alonso del Arte, Feb 21 2012

From Peter Munn, Aug 09 2019: (Start)

The factorization pattern of a number's divisors (as defined in A191743) determines whether a number is a term.

There are no semiprimes in the sequence, and a 3-almost prime is present if and only if its largest prime factor is less than its square root. The first term that is a 4-almost prime is 220.

The effect of this sieve can be compared against the A270877 trapezoidal sieve. Each unmarked number k marks k-1 numbers in both sieves; but the largest number marked by k in this sieve is k^2, about twice the largest number marked by k in A270877 (the triangular number T_k = k(k+1)/2). The relative densities early in the two sequences are illustrated by a(10) = 18 < A270877(10) = 19, a(100) = 312 > A270877(100) = 268, a(1000) = 4297 > A270877(1000) = 2894. (End)

The sequence is generated by a sieving method with iterated selection of intervals of the natural numbers as if they were forming a chain of contact points on which a stone could re-bounce once launched at some specific position at the small numbers.

Image a stone with an initial kinetic energy t, which is diminished/dissipated by 1 unit each time it rebounds from the "water surface" of the residual sequence; it rebounds t times and sinks once it has slowed down to t=1. The numbers underneath the arcs of this flight, but not the contact points, are eliminated. We look at the limit of repeatedly skipping stones each time starting at new launching points with larger initial t. In detail:

Start with the set of natural numbers. Let a(0)= t define t. Jump t positions to the right, erase t positions; from the last erased position jump t-1 positions to the right, erase t-1 positions; ...; jump 1 position to the right, erase 1 position. Go to the smallest i>t. Set t=i. Repeat.

Stone skipping sequences are a generalized case of scarce sequences; see A137292.