A278577 Ramanujan function tau(p) as p runs through the prime powers: a(n) = A000594(A000961(n)).

1, -24, 252, -1472, 4830, -16744, 84480, -113643, 534612, -577738, 987136, -6905934, 10661420, 18643272, -25499225, -73279080, 128406630, -52843168, -196706304, -182213314, 308120442, -17125708, 2687348496, -1696965207, -1596055698, -5189203740, 6956478662, 2699296768, -15481826884, 9791485272

Offset: 1

Author

N. J. A. Sloane, Nov 29 2016

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1933 from Daniel Suteu)
• D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433.
• D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]

Crossrefs

Cf. A000594, A000961, A076847.

Programs

Mathematica

Join[{1}, RamanujanTau[Select[Range[100], PrimePowerQ]]] (* Paolo Xausa, May 11 2024 *)

PARI

list(lim) = apply(ramanujantau, select(x -> x == 1 || isprimepower(x), vector(lim, i, i))); \\ Amiram Eldar, Jan 09 2025

Python

from itertools import count, islice
from sympy import primefactors, divisor_sigma
def A278577_gen(): # generator of terms
    yield 1
    for n in count(2):
        f = primefactors(n)
        if len(f) == 1:
            p, m = f[0], n+1>>1
            yield (q:=n**4)*(p*n-1)//(p-1)-24*((0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*(m*divisor_sigma(m))**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + q)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m)))
A278577_list = list(islice(A278577_gen(),10)) # Chai Wah Wu, Nov 11 2022