A337032 a(n) = (n*sigma_9(n) - tau(n))/7 = (A282254(n) - A000594(n))/7, where tau is Ramanujan's tau, sigma_9(n) = Sum_{d divides n} d^9.

0, 150, 8400, 150300, 1394400, 8656200, 40356000, 153679800, 498153600, 1431378900, 3705270000, 8863150800, 19694152800, 41402744400, 82382680800, 157380332400, 288000115200, 511088547150, 875865085200, 1465721632200, 2382961862400, 3801687211800, 5918070367200, 9075809181600

Offset: 1

Author

Jianing Song, Aug 12 2020

Keywords

• nonn

Comments

D. H. Lehmer shows that tau(n) == n*sigma_9(n) (mod 7), so a(n) is an integer for all n. Furthermore, if n == 3, 5, 6 (mod 7) then tau(n) == n*sigma_9(n) (mod 49). See the Wikipedia link below. It seems that the latter congruence also holds for most of the other numbers. Among the 571 numbers in [1, 1000] congruent to 0, 1, 2, 4 modulo 7, tau(n) == n*sigma_9(n) holds for 311 n's, and among the 5715 numbers in [1, 10000] congruent to 0, 1, 2, 4 modulo 7, the congruence holds for 3492 n's.

It seems that 150 divides a(n) for all n. There are no counterexamples for n <= 10000.

Number of n's in [2, N] which satisfy the higher-order congruence tau(n) == n*sigma_9(n) (mod 7^e) but not tau(n) == n*sigma_9(n) (mod 7^(e+1)):

N = 1000:

e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total

---+----------------------+-------------------------+-------

1 | 0 | 260 | 260

---+----------------------+-------------------------+-------

2 | 358 | 80 | 438

---+----------------------+-------------------------+-------

3 | 45 | 195 | 240

---+----------------------+-------------------------+-------

4 | 24 | 28 | 52

---+----------------------+-------------------------+-------

5 | 2 | 5 | 7

---+----------------------+-------------------------+-------

6 | 0 | 2* | 2

* n = 686, 942.

N = 10000:

e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total

---+----------------------+-------------------------+-------

1 | 0 | 2223 | 2223

---+----------------------+-------------------------+-------

2 | 3368 | 728 | 4096

---+----------------------+-------------------------+-------

3 | 466 | 2280 | 2746

---+----------------------+-------------------------+-------

4 | 397 | 384 | 781

---+----------------------+-------------------------+-------

5 | 46 | 87 | 133

---+----------------------+-------------------------+-------

6 | 6 | 12 | 18

---+----------------------+-------------------------+-------

7 | 2** | 0 | 2

** n = 5185, 9021.

Examples

			a(2) = (n*sigma_9(2) - tau(2))/7 = (2*(1^9+2^9) - (-24))/7 = 1050/7 = 150;
a(3) = (n*sigma_9(3) - tau(3))/7 = (3*(1^9+3^9) - 252)/7 = 58800/7 = 8400.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Wikipedia, Congruences for the tau function.

Crossrefs

Cf. A000594, A282254, A027860.

Programs

Mathematica

a[n_] := (n * DivisorSigma[9, n] - RamanujanTau[n]) / 7; Array[a, 24] (* Amiram Eldar, Jan 10 2025 *)

PARI

a(n) = (n*sigma(n, 9) - polcoeff( x * eta(x + x * O(x^n))^24, n))/7;