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Divisors of numbers and their continuations in the theory of partitions. (English) JFM 47.0117.01

Unter \(a_{n,k}\) versteht der Verf. die Summe \(\sum s_1 s_2 \dots s_k,\) erstreckt über sämtliche Lösungen der diophantischen Gleichung \[ s_1 m_1 + s_2 m_2 + \cdots + s_k m_k = n, \] \(s_\nu, m_\nu\) positiv ganz und die \(m_\nu\) voneinander verschieden. Es wird für \[ A_k=\sum_{n=1}^\infty a_{n,k}q^n \] die folgende symbolische Darstellung bewiesen: \[ 2^{2k}(2k+1)! A_k = (-1)^k \frac {1}{J_1} J(J^2 -1^2)(J^2 - 3^2) \dots (J^2 - (2k -1)^2), \] in welcher nach Entwicklung \(J^r\) durch \[ J_r =1- 3^rq + 5^rq^3 - 7^rq^6 + \cdots, \] mit den Trigonalzahlen als Exponenten von \(q\) zu ersetzen ist. Ähnliche Darstellungen gelten für die weiteren vom Verf. ausführlich untersuchten Reihen \(B, C, D, E, F; B\) ist ähnlich definiert wie \(A = A_k,\) mit dem Unterschied, daßan Stelle von \(a_{n,k}\) \[ b_{n,k} = \sum (- 1)^{s_1 +s_2 +\cdots +s_k +k}s_1s_2\dots s_k \] tritt. Bei den Reihen \(C, D\) werden nur solche Zerfällungen von \(n\) zugelassen, bei denen sämtliche \(m_\nu\) ungerade sind, bei \(E, F\) solche, die von der Form \(5r\pm 1,\) bei \(G, H\) solche, die von der Form \(5r\pm 2\) sind.

MSC:

11P81 Elementary theory of partitions
11D04 Linear Diophantine equations
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Online Encyclopedia of Integer Sequences:

d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.
a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).
MacMahon’s generalized sum of divisors function.
MacMahon’s generalized sum of divisors function.
Generalized sum of divisors function: excess of sum of odd divisors of n over sum of even divisors of n.
Generalized sum of divisors function.
Sum of divisors d of n such that n/d is odd.
Generalized sum of divisors function.
Number of partitions of n with exactly two part sizes.
Generalized divisor function. Partitions of n using only 3 types of piles.
Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows.
Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.
Number of odd divisors of n minus number of even divisors of n.
Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.
Triangle of generalized sum of divisors function, read by rows.
Generalized sum of divisors function: third diagonal of A060044.
Generalized sum of divisors function: third diagonal of A060047.
Triangle of generalized sum of divisors function, read by rows.
Triangle of generalized sum of divisors function, read by rows.
Triangle of generalized sum of divisors function, read by rows.
Generalized sum of divisors function: second diagonal of A060184.
Generalized sum of divisors function: third diagonal of A060184.
Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).
A column and diagonal of A060187.
A column and diagonal of A060187 (k=3).
A column and diagonal of A060187 (k=4).
Numbers n such that tau(n^2+1) - tau(n^2) = 1 where the function tau(n) is the number of positive divisors of n.
Numbers k such that tau(sigma(k)) = sopf(k).
Numbers n such that sigma(tau(n)) equals the sum of distinct primes dividing n.
Numbers k such that phi(tau(k)) = sopf(k).
Numbers k such that tau(phi(k))= sopf(k).
Numbers k such that phi(tau(k)) = tau(sopf(k)).
Numbers k such that tau(phi(k)) = phi(sum-of-prime-divisors(k)).
Numbers k such that tau(phi(k)) = sigma(sopf(k)).
Numbers n such that tau(tau(n)) = sopf(sopf(n)), sopf = A008472.
Numbers n such that phi(tau(n))= rad(n)