cp's OEIS Frontend

Or, in the list of natural numbers (A000027), replace n with its divisors.

This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.

Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008

Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011

{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012

Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012

Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013

The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015

The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017

T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019

Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025

All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

Alternating row sums of A056538. - Omar E. Pol, Feb 17 2024

Does a constant analogous to the median abundancy index (see A353617) exist for this function? Particularly, does a constant exist such that the numbers having the value a(n)/n greater than this constant have natural density exactly 1/2? Using the first 10^7 values, one observes that if it exists, this constant appears to converge around 0.726. - Shreyansh Jaiswal, Apr 16 2025

Binary expansion of n consists of single 1's diluted by (possibly empty) equal-sized blocks of 0's.

According to Stolarsky's Theorem 2.1, all numbers in this sequence are sturdy numbers; this sequence is a subsequence of A125121. - T. D. Noe, Jul 21 2008

These are the numbers k > 0 for which k + 2^m = k*2^n + 1 has a solution m,n > 0. For k > 1, these are numbers k such that (k - 2^x)*2^y + 1 = k has a solution in positive integers x,y. In other words, (k - 1)/(k - 2^x) = 2^y for some x,y > 0. If t = (2^m - 1)/(2^n - 1) is a term of this sequence (i.e. if and only if n|m), then t' = t + 2^m = t*2^n + 1 is also a term. Primes in this sequence (A245730) include: all Mersenne primes (A000668), all Fermat primes (A019434), and other primes (73, 262657, 4432676798593, ...). - Thomas Ordowski, Feb 14 2024

The union of N, 2N, 3N, ..., where N = {1, 2, 3, 4, 5, 6, ...}. In other words, the numbers {m*n, m >= 1, n >= 1} sorted into nondecreasing order.

Considering the maximal rectangle in each of the Ferrers graphs of partitions of n, a(n) is the smallest such maximal rectangle; a(n) is also an inverse of A006218. - Henry Bottomley, Mar 11 2002

The numbers in A003991 arranged in numerical order. - Matthew Vandermast, Feb 28 2003

Least k such that tau(1) + tau(2) + tau(3) + ... + tau(k) >= n. - Michel Lagneau, Jan 04 2012

The number 1 appears only once, primes appear twice, squares of primes appear thrice. All other positive integers appear at least four times. - Alonso del Arte, Nov 24 2013

Row n has length sigma(n) = A000203(n).

Row sums give n*A000005(n) = A038040(n).

Column 1 is A000027.

Both columns 2 and 3 are A032742, n > 1.

For any k > 0 and t > 0, the sequence contains exactly one run of k consecutive t's. - Rémy Sigrist, Feb 11 2019

From Omar E. Pol, Dec 04 2019: (Start)

The number of parts congruent to 0 (mod m) in row m*n equals sigma(n) = A000203(n).

The number of parts greater than 1 in row n equals A001065(n), the sum of aliquot parts of n.

The number of parts greater than 1 and less than n in row n equals A048050(n), the sum of divisors of n except for 1 and n.

The number of partitions in row n equals A000005(n), the number of divisors of n.

The number of partitions in row n with an odd number of parts equals A001227(n).

The sum of odd parts in row n equals the sum of parts of the partitions in row n that have an odd number of parts, and equals the sum of all parts in the partitions of n into consecutive parts, and equals A245579(n) = n*A001227(n).

The decreasing records in row n give the n-th row of A056538.

Row n has n 1's which are all at the end of the row.

First n rows contain A000217(n) 1's.

The number of k's in row n is A126988(n,k).

The number of odd parts in row n is A002131(n).

The k-th block in row n has A027750(n,k) parts.

Right border gives A000012. (End)

The r-th row of the triangle begins at index k = A160664(r-1). - Samuel Harkness, Jun 21 2022

First differs from A273103 at a(14). - Omar E. Pol, May 15 2016

Positive values of triangle A168016.

The number of terms of row n is equal to the number of divisors of n: A000005(n).

Note that the last term of each row is the number of partitions of n: A000041(n).

Also, it appears that row n lists the partition numbers of the divisors of n. [Omar E. Pol, Nov 23 2009]

Consider the diagram with overlapping periodic curves that appears in the Links section (figure 1). The number of curves that contain the point [n,0] equals the number of divisors of n. The simpler interpretation of the diagram is that the curve of diameter d represents the divisor d of n. Now here we introduce a new interpretation: the curve of diameter d that contains the point [n,0] represents the divisor c of n, where c = n/d. This version of the model has the property that each odd quadrant centered at [n,0] contains the curves that represent the even divisors of n, and each even quadrant centered at [n,0] contains the curves that represent the odd divisors of n.

We can find the n-th row of the triangle as follows:

Consider only the semicircumferences that contain the point [n,0].

In the second quadrant from bottom to top we can see the curves that represent the odd divisors of n in decreasing order. Also we can see these curves in the fourth quadrant from top to bottom.

Then, if n is an even number, in the first quadrant from top to bottom we can see the curves that represent the even divisors of n in increasing order. Also we can see these curves in the third quadrant from bottom to top (see example).

Sequences of the same family are shown below:

-----------------------------------

Triangle Order of divisors of n

-----------------------------------

This seq. odd v t.w. even ^

A299483 odd ^ t.w. even v

A319844 even v t.w. odd ^

A319845 even ^ t.w. odd v

A319846 odd v t.w. even v

A319847 odd ^ t.w. even ^

A319848 even v t.w. odd v

A319849 even ^ t.w. odd ^

-----------------------------------

In the above table we have that:

"even v" means "even divisors of n in decreasing order".

"even ^" means "even divisors of n in increasing order".

"odd v" means "odd divisors of n in decreasing order".

"odd ^" means "odd divisors of n in increasing order".

"t.w." means "together with".