cp's OEIS Frontend

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

%I A278577 #30 Jan 09 2025 02:05:27
%S A278577 1,-24,252,-1472,4830,-16744,84480,-113643,534612,-577738,987136,
%T A278577 -6905934,10661420,18643272,-25499225,-73279080,128406630,-52843168,
%U A278577 -196706304,-182213314,308120442,-17125708,2687348496,-1696965207,-1596055698,-5189203740,6956478662,2699296768,-15481826884,9791485272
%N A278577 Ramanujan function tau(p) as p runs through the prime powers: a(n) = A000594(A000961(n)).
%H A278577 Seiichi Manyama, <a href="/A278577/b278577.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1933 from Daniel Suteu)
%H A278577 D. H. Lehmer, <a href="http://dx.doi.org/10.1215/S0012-7094-47-01436-1">The Vanishing of Ramanujan's Function tau(n)</a>, Duke Mathematical Journal, 14 (1947), pp. 429-433.
%H A278577 D. H. Lehmer, <a href="/A000594/a000594.pdf">The Vanishing of Ramanujan's Function tau(n)</a>, Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
%t A278577 Join[{1}, RamanujanTau[Select[Range[100], PrimePowerQ]]] (* _Paolo Xausa_, May 11 2024 *)
%o A278577 (Python)
%o A278577 from itertools import count, islice
%o A278577 from sympy import primefactors, divisor_sigma
%o A278577 def A278577_gen(): # generator of terms
%o A278577     yield 1
%o A278577     for n in count(2):
%o A278577         f = primefactors(n)
%o A278577         if len(f) == 1:
%o A278577             p, m = f[0], n+1>>1
%o A278577             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)))
%o A278577 A278577_list = list(islice(A278577_gen(),10)) # _Chai Wah Wu_, Nov 11 2022
%o A278577 (PARI) list(lim) = apply(ramanujantau, select(x -> x == 1 || isprimepower(x), vector(lim, i, i))); \\ _Amiram Eldar_, Jan 09 2025
%Y A278577 Cf. A000594, A000961, A076847.
%K A278577 sign
%O A278577 1,2
%A A278577 _N. J. A. Sloane_, Nov 29 2016