cp's OEIS Frontend

To clarify our terminology: We say T is the convolution triangle of a (or T is the partition transform of a), if a is a sequence of integers defined for n >= 1, and the transformation, as defined by the Maple function below, applied to a, returns T. In the generated lower triangular matrix T, i.e., in the convolution triangle of a, T(n, 1) = a(n) for n >= 1.

For instance, let a(n) = Bell(n), then we call A357583 the convolution triangle of the Bell numbers, but not A205574, which is also called the "Bell convolution triangle" in the comments but leads to a different triangle. Similarly, if a(n) = n!, then A090238 is the convolution triangle of the factorial numbers, not A084938. Third example: A128899 is the convolution triangle of the Catalan numbers in our setup, not A059365. More generally, we recommend that when computing the transform of a 0-based sequence to use only the terms for n >= 1 and not to shift the sequence.

Note that T is (0, 0)-based and the first column of a convolution triangle always is 1, 0, 0, 0, ... and the main diagonal is 1, 1, 1, 1, ... if a(1) = 1. The (1, 1)-based subtriangle of a genuine convolution triangle, i.e., a convolution triangle without column 0 and row 0, is often used in place of the convolution triangle but does not fit well into some applications of the convolution triangles.

As in A166062, the offset is rather arbitrary.

The sequence contains numbers like 210 which are not in A006954.

One could also consider dividing by the largest prime divisor of A027642 instead of A089026, which yields 1, 1, 2, 1, 6, 1, 6, 1, 6, 1, 6, 1, 210, 1, 2, 1, 30, 1, 42, 1, 30, ... as an alternative version.

These are the Clausen numbers based on the proper divisors of n whereas the classical Clausen numbers A160014 are based on all divisors of n. (The proper divisors are the divisors of n that are less than n.) - Peter Luschny, Aug 20 2022

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. (The prime marker sequence A089026 is defined by p(n)=n if n is prime and p(n)=1 otherwise.)

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194184 is a permutation of the positive integers, with inverse A195185. (The prime marker sequence A089026 is given by p(n)=n if n is prime and p(n)=1 otherwise).

Same as A061006 except for a(4) = 0 (Wilson's Theorem). - Georg Fischer, Oct 12 2018

Also nonprime n such that sigma(n)*phi(n) > (n-1)^2. - Benoit Cloitre, Apr 12 2002

If p is a term of the sequence, then the index n for which a(n) = p is given by n := b(p) := 1 + Sum_{k>=2} PrimePi(p^(1/k)). Here, the sum has floor(log_2(p)) positive terms. For any m > 0, the greatest number n such that a(n) <= m is also given by b(m), thus, b(m) is the number of such prime powers <= m. - Hieronymus Fischer, May 31 2013

That 8 and 9 are the only two consecutive integers in this sequence is known as Catalan's Conjecture and was proved in 2002 by Preda Mihăilescu. - Geoffrey Critzer, Nov 15 2015

By Wolstenholme's theorem, p divides a(p-1) for prime p > 3. - T. D. Noe, Sep 05 2002

Also p divides a( (p-1)/2 ) for prime p > 3. - Alexander Adamchuk, Jun 07 2006

The rationals a(n)/A007407(n) converge to Zeta(2) = (Pi^2)/6 = 1.6449340668... (see the decimal expansion A013661).

For the rationals a(n)/A007407(n), n >= 1, see the W. Lang link under A103345 (case k=2).

See the Wolfdieter Lang link under A103345 on Zeta(k, n) with the rationals for k=1..10, g.f.s and polygamma formulas. - Wolfdieter Lang, Dec 03 2013

Denominator of the harmonic mean of the first n squares. - Colin Barker, Nov 13 2014

Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). Checked up to n = 10^5. - Amiram Eldar and Thomas Ordowski, Jul 28 2019

True if n is prime, by Wolstenholme's theorem. It remains to show that gcd(n, a(n-1)) = 1 if n > 3 is composite. - Jonathan Sondow, Jul 29 2019

From Peter Bala, Feb 16 2022: (Start)

Sum_{k = 1..n} 1/k^2 = 1 + (1 - 1/2^2)*(n-1)/(n+1) - (1/2^2 - 1/3^2)*(n-1)*(n-2)/((n+1)*(n+2)) + (1/3^2 - 1/4^2)*(n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) - (1/4^2 - 1/5^2)*(n-1)*(n-2)*(n-3)*(n-4)/((n+1)*(n+2)*(n+3)*(n+4)) + .... Cf. A082687 and A120778.

This identity allows us to extend the definition of Sum_{k = 1..n} 1/k^2 to non-integral values of n. (End)

Numerators of the Eulerian numbers T(-2,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - Michael J. Collins, Oct 10 2024

Rows equal columns and are periodic. No zero elements are found (conjecture). The recurrence is related to the recurrence for the Mahonian numbers. The main diagonal is the Dirichlet inverse of the Euler totient function A023900 (conjecture). [The 2nd and 3rd formulas state that the conjecture is correct. R. J. Mathar, Sep 16 2017]

The sums from n=1 to infinity of T(n,k)/n converge to the Mangoldt function for column k (conjecture).

If gcd(n,k)=1 then T(n,k)=1 and if T(n,k)=1 then gcd(n,k)=1 (conjecture).

The Dirichlet generating functions for s > 1 for the columns appear to be (see A054535):

Zeta(s)*(1 + (Sum over all possible combinations of products of negative distinct prime factors of k, up to rearrangement, 1/((-1* first distinct prime factor)*(-1*second distinct prime factor)*(-1*third distinct prime factor * ...))^(s-1))).

Examples:

k=1: Zeta(s)

k=2: Zeta(s)*(1 - 1/2^(s-1))

k=3: Zeta(s)*(1 - 1/3^(s-1))

k=4: Zeta(s)*(1 - 1/2^(s-1))

k=5: Zeta(s)*(1 - 1/5^(s-1))

k=6: Zeta(s)*(1 - 1/2^(s-1) - 1/3^(s-1) + 1/6^(s-1))

k=7: Zeta(s)*(1 - 1/7^(s-1))

...

k=30: Zeta(s)*(1 - 1/2^(s-1) - 1/3^(s-1) - 1/5^(s-1) + 1/6^(s-1) + 1/10^(s-1) + 1/15^(s-1) - 1/30^(s-1))

...

(conjecture)

See triangle A142971 for negative distinct prime factors.

This could probably be checked by matrix multiplication.

The signs of the eigenvalues of this matrix are a rearrangement of the Mobius function A008683 (conjecture). The first few eigenvalues are:

{1.0000}

{-1.4142, 1.4142}

{-2.6554, 1.8662, -1.2108}

{-3.4393, 2.1004, -1.6611, 0}

{-4.7711, -3.3867, 2.5910, -1.4332, 0}

{-5.2439, -3.4641, 3.4641, 2.5169, -2.2730, 0}

The relation to Dirichlet characters for the entries in this matrix appears appears to be formulated in terms of the sequence A089026 which is equal to n if n is a prime, otherwise equal to 1. See Mathematica program below. [Mats Granvik, Nov 23 2013]

From Mats Granvik, Jun 19 2016: (Start)

Remark about the Dirichlet generating function for the whole matrix: Subtracting the first column (in the form of zeta(c)) of the matrix gives us the limit: lim_{c->1} zeta(s)*zeta(c)/zeta(c+s-1)-zeta(c) = -zeta'(s)/zeta(s) which is the classical Dirichlet generating function for the von Mangoldt function.

For n >= k, see A231425, this matrix has row sums equal to zero except for the first row:

1=1

1-1=0

1+1-2=0

1-1+1-1=0

1+1+1+1-4=0

...

log(A014963(n)) = Sum_{k>=1} A191898(n,k)/k, for n>1.

log(A014963(k)) = Sum_{n>=1} A191898(n,k)/n, for k>1.

log(A014963(n)) = limit of zeta(s)*(Sum_{d divides n} A008683(d)/d^(s-1)) as s->1, for n>1.

A008683(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n.

A008683(n) = Sum_{n=1..k} A191898(n,k)*exp(-i*2*Pi*n/k)/k.

(End)

From Mats Granvik, Aug 09 2016: (Start)

The Dirichlet generating function for the matrix is zero for any pair c and s = 2 - c and any pair s and c = 2 - s except at the pole c = 1 and s = 1 where it is indeterminate.

In the Mathematica program section, in the expression for the matrix as Dirichlets characters, the variables s and c can apparently be any pair of positive integers.

Limits related to the Dirichlet generating function for the matrix: Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s-1) = ZetaZero(n). Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s*c-1) = ZetaZero(n)/(1+ZetaZero(n)).

(End)